Variational methods for problems from plasticity theory and for generalized Newtonian 掳uids

285

Transcript of Variational methods for problems from plasticity theory and for generalized Newtonian 掳uids

Max�Planck�Institut

f�ur Mathematik

in den Naturwissenschaften

Leipzig

Variational methods for problems from

plasticity theory and for generalized

Newtonian �uids

by

Martin Fuchs and Gregory Seregin

Lecture notes no�� � ����

Variational Methods For Problems From

Plasticity Theory And For Generalized

Newtonian Fluids

Martin Fuchs Gregory Seregin

Contents

Introduction �

� Weak solutions to boundary value problemsin the deformation theory of perfect elastoplasticity �

��� Preliminaries � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

��� The classical boundary value problem for the equilibrium stateof a perfect elastoplastic body and its primary functionalformulation � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

��� Relaxation of convex variational problems in non re�exive spaces�General construction � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

��� Weak solutions to variational problems of perfectelastoplasticity � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� Di�erentiability properties of weak solutions to boundary valueproblems in the deformation theory of plasticity ��

��� Preliminaries � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

��� Formulation of the main results � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

��� Approximation and proof of Lemma ����� � � � � � � � � � � � � � � � � � � � � � �

��� Proof of Theorem ����� and a local estimate ofCaccioppoli type for the stress tensor � � � � � � � � � � � � � � � � � � � � � � � � � �

��� Estimates for certain systems of PDE�s with constant coe�cients���

�� The main lemma and its iteration � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

�� Proof of Theorem ����� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

�� Open Problems � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ���

��� Remarks on the regularity of minimizers of variationalfunctionals from the deformation theory of plasticitywith power hardening � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ���

Variational Methods �

Appendix A ���

A�� Density of smooth functions in spaces of tensor�valued functions��

A�� Density of smooth functions in spacesof vector�valued functions � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ���

A�� Some properties of the space BD ���Rn� � � � � � � � � � � � � � � � � � � � � ���

A�� Jensen�s Inequality � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� Quasi�static �uids of generalized Newtonian type ���

��� Preliminaries � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ���

��� Partial C� regularity in the variational setting � � � � � � � � � � � � � � � � �

��� Local boundedness of the strain velocity � � � � � � � � � � � � � � � � � � � � � � ���

��� The two�dimensional case � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ���

��� The Bingham variational inequalityin dimensions two and three � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ���

�� Some open problems and comments concerning extensions � � � � ���

� Fluids of PrandtlEyring type and plastic materialswith logarithmic hardening law ���

��� Preliminaries � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ���

��� Some function spacesrelated to the Prandtl�Eyring �uid model � � � � � � � � � � � � � � � � � � � � ���

��� Existence of higher order weak derivatives and aCaccioppoli�type inequality � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ���

��� Blow�up� the proof of Theorem ����� for n � � � � � � � � � � � � � � � � � ��

��� The two�dimensional case � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

�� Partial regularity for plastic materials with logarithmichardening � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ���

�� A general class of constitutive relations � � � � � � � � � � � � � � � � � � � � � � ��

Appendix B ��

B�� Density results � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

Bibliography ��

Introduction

In this monograph we develop a rigorous mathematical analysis of variationalproblems describing the equilibrium con�guration for certain classes of solidsand also for the stationary �ow of some incompressible generalized Newtonian�uids� However� even if we restrict ourselves to the time�independent setting�it is not possible to include all aspects which might be of interest� So we madea selection which has been in�uenced by our scienti�c activities over the lastyears� In particular� we concentrate on variational problems from the defor�mation theory of plasticity and on �uid models whose stress�strain relationcan be formulated in terms of a dissipative potential� The mathematical formof both problems is very close and may be reduced to the study of variationalintegrals with convex integrands depending only on the symmetric part of thegradient of the unknown vector�valued functions� In the applications thesefunctions represent either displacement �elds of a body or velocity �elds ofincompressible �ows�

There are many books devoted to the correct mathematical formulation ofproblems from the mechanics of continua and �uids and which in�uenced usin various ways� Without being complete let us mention the monographs ofG� Duvaut and J�L� Lions �DL�� O�A� Ladyzhenskaya �L��� P�P� Mosolov andV�P� Mjasnikov �MM��� J� Nec�as and I� Hlav�a�cek �NH�� E� Zeidler �Z�� G� As�tarita and G� Marrucci �AM�� R� Bird� R� Armstrong and O� Hassager �BAH��and I� Ionescu and M� Sofonea �IS�� After presenting some physical backgroundfor the problems under consideration these authors introduce appropriate con�cepts of weak solutions and prove existence theorems in spaces of generalizedfunctions� From the point of view of applications the most interesting problemconcerns the smoothness properties of these generalized solutions which willbe the main issue of our book� But let us �rst brie�y comment the approach

Variational Methods

towards existence� by convexity� Lp�Korn�s inequality and additional quitenatural conditions concerning the data� the so�called direct method from thecalculus of variations gives existence in some Sobolev space provided the in�tegrand is of superlinear growth� We therefore obtain existence for plasticitywith power hardening studied in chapter �� section �� and also for the genera�lized Newtonian �uids investigated in chapter �� However� for integrands withlinear growth as they occur in perfect plasticity� the problem of �nding suitableclasses of admissible deformations is not so trivial� The right space consists ofall functions having bounded deformation� a de�nition and further discussioncan be found in the works of G� Anzellotti and M� Giaquinta �AG��� H� Ma�thies� G� Strang and E� Christiansen �MSC�� P� Suquet �Su� and R� Temamand G� Strang �ST��� Existence of weak solutions in this space was proved byG� Anzellotti and M� Giaquinta �AG��� R� Kohn and R� Temam �KT�� R� Hardtand D� Kinderlehrer �HK�� R� Temam �T� and G� Seregin �Se��� In the presentbook we use the approach of the second author outlined in �Se�� which is closeto the one of R� Temam and G� Strang �ST��� Roughly speaking� we concen�trate on a dual variational problem for the stress tensor which in our opiniongives more chances for investigating di�erentiability properties and moreoverit has a clear physical meaning� Chapter � contains a general scheme for therelaxation of convex variational problems being coercive on some non�re�exivespaces like L� or W �

� and as an application we will give a detailed descripti�on of our approach towards perfect plasticity� Besides plasticity with powerhardening and perfect plasticity there is an other model which in a certainsense lies in between these two cases and which is known as plasticity withlogarithmic hardening� Here the integrand is not of power growth for somep � � which makes it necessary to introduce �new� function spaces in whichsolutions should be located� Integrands of logarithmic type also occur in thestudy of Prandtl�Eyring �uids� For this reason we decided to present a uni��ed treatment of plasticity with logarithmic hardening and of Prandtl�Eyring�uids in chapter ��

As already mentioned our main concern is the analysis of regularity of weaksolutions� In chapter � we adress this question for perfect elastoplasticity inclu�ding the Hencky�Il�yushin model as a special case� According to our knowledgethe problem of regularity for integrands with linear growth and for vector�valued functions is discussed only in the paper �AG�� of G� Anzellotti andM� Giaquinta where they prove partial C��regularity� i�e� smoothness up toa relatively closed set with vanishing Lebesgue measure� assuming in additionthat the integrand is strictly convex which is not the case for Hencky�Il�yushinplasticity� Our approach towards regularity is completely di�erent and based

FUCHS et al�

on the idea that the stress tensor is � from the physical point of view � themost important quantity since for example it determines the elastic and pla�stic zones within the body� We therefore study the dual variational problemfor the stress tensor which has a unique solution but the form of this problemdi�ers from the setting in standard variational calculus� the functional doesnot involve any derivatives of the unknown functions� the yield condition actsas a pointwise constraint and the equilibrium equations for the stresses haveto be incorporated in the admissible class� Despite of these di�culties we ob�tain additional regularity for the stress tensor and with the help of the dualityrelations which can be regarded as a weak form of the constitutive equationswe also establish some regularity for the displacement �eld�

In chapter � we study the local regularity of solutions to variational problemsdescribing the stationary �ow of generalized Newtonian �uids including alsoviscoplastic �uids of Bingham type� The Bingham case and its extension in�herits the main di�culties since the dissipative potential is not smooth at theorigin� Nevertheless we get partial regularity of the strain velocity on the com�plement of the rigid zone� Other results for the Bingham model concern theboundedness of the strain velocity and� for two�dimensional problems� justcontinuity of this tensor� For other classes of generalized Newtonian �uidswhose dissipative potential is obtained via a smooth lower order perturbationof a quadratic leading part we prove �partial� regularity in the usual sense�During our investigations we �rst assume that the velocity is also small inorder to get a minimisation problem� If we drop this assumption� then thevariational problem has to be replaced either by a variational inequality or bysome elliptic system which di�ers from the time�independent Navier�Stokessystem since its leading part is nonlinear� By introducing some arti�cial newvolume load we can reduce this situation to the one studied before by the wayobtaining the same regularity results at least in dimensions two and three� Itshould be noted that all our investigations are limited to incompressible �owswhich means that we have to consider �elds with vanishing divergence�

Chapter � presents the regularity theory for the logarithmic case working witha natural notion of weak solutions� For plasticity with logarithmic hardeningwe directly investigate the minimizing deformation �eld proving partial regu�larity in three dimensions and global smoothness in the two�dimensional case�From this and the constituent equations we deduce corresponding results forthe stress tensor� As to the Prandtl�Eyring �uid model exactly the same smoo�thness theorems hold provided we assume smallness of the velocity �eld� It isan open problem if this restriction can be removed�

Variational Methods

In the end� let us point out that chapters �� � and �� although devoted to dif�ferent subjects� are strongly connected by the methods we use� For example�we apply suitable regularisations to get variational problems which are easierto handle� we prove weak di�erentiability of stresses or strain velocities� andwe use blow�up arguments for establishing �partial� regularity� For keepingthe exposition selfcontained we decided to include some background materialconcerning function spaces� The reader will �nd the necessary information atthe end of chapter � and chapter ��

Last but not least we want to thank Mrs� C� Peters for doing an excellenttyping job� The second author also acknowledges support from INTAS� grantNo� �����

Chapter �

Weak solutions to boundary

value problems in the

deformation theory of perfect

elastoplasticity

��� Preliminaries

Chapter � is organized as follows� In the �rst section we discuss the classicalboundary value problem for the equilibrium con�guration of a perfect ela�stoplastic body within the framework of the so�called deformation theory ofplasticity and give some preliminary version of a functional formulation whichtakes the form of a minimax problem� At this stage we impose the conditionthat the strain tensor is a summable function� We then show that the mini�max problem is equivalent to two variational problems for the displacement�eld and the stress tensor� respectively� But the minimisation problem for thedisplacement �eld is in general not solvable if we require summability of thestrain tensor which is a consequence of considering some concrete examples�In section two we therefore describe an abstract procedure providing a suitablerelaxation of variational problems being coercive on non�re�exive classes likeL� orW �

� � The main section of chapter � is section three in which we apply theabstract scheme to concrete problems of perfect elastoplasticity� As a resultwe obtain a relaxation of the above mentioned minimax problem with corre�sponding weak solutions� i�e�� the solutions of the relaxed minimax problemare� in some natural sense� generalized solutions of the original minimax pro�blem� Moreover� this approach immediately leads to a relaxed version of the

Variational Methods �

minimisation problem for the displacement �eld� the correct class for the ad�missible displacement �elds turns out to be the space of functions of boundeddeformation where the strain tensor is just a bounded measure�

�� FUCHS et al�

��� The classical boundary value problem for

the equilibrium state of a perfect elasto�

plastic body and its primary functional

formulation

Let M n�nsym be the space of all symmetric matrices of order n� We will use the

following notation

u � v � ui vi� juj � pu � u�

u� v � �uivj�� u� v � ���u� v � v � u� for u � �ui�� v � �vi� � Rn �

� � � � �ij�ij� j�j � p� � ��

�� � ��ij�j� � Rn for � � ��ij�� � � ��ij��

� � ��ij� � M n�nsym � and � � ��i� � Rn �

where the convention of summation over Latin indices running from � to n isadopted�

We consider an elastoplastic body whose undeformed state is represented bya set � � R

n � n � �� �� It is assumed that � is a bounded Lipschitz domainwith boundary consisting of two measurable parts ��� and ��� satisfying theconditions �� � ������� and ������� � �� Under the action of given forcesthe body is deformed� In mathematical terms this process is described by thedisplacement �eld � � x u�x� � R

n and the tensor of small deformations�strain tensor� which is the symmetric part of the gradient of the vector�valued�eld u� i�e�

��u� � ��ij�u��� ��ij�u� � ui�j � uj�i� ui�j � �ui��xj�

i� j � �� �� � � � � n�

Variational Methods ��

The strain tensor contains all the important information about the geometryof the deformation whereas the stress state of the elastoplastic body is charac�terized by the so�called stress tenssor � � � M

n�nsym �

Within the framework of the deformation theory of plasticity the classicalboundary value problem modelling the static equilibrium con�guration of thegiven elastoplastic body can be formulated as follows �see� for instance� �Kl���We look for the displacement �eld u � � R

n and the stress tensor � � � M

n�nsym which satisfy the following three groups of relations�

Equilibrium equations for the stresses

div ��x� � f�x� � �� x � �� ��x���x� � F �x�� x � ���������

Deformation relations

���u� � �ui�j � uj�i� on �� u � u� on ���������

Constitutive equations

�������������������

��u�x�� � �e�x� � �p�x�� x � �� F���x�� � �� x � ��

�e�x� ��

n�K��tr ��x� � �

���D�x�� x � ��

�p�x� � �� � ��x�� � � for any � � M n�nsym s�t� F��� � �

and for all x � ��

�����

Here div � � ��ij�j� is the divergence of the tensor �� f � � Rn � F � ���

Rn are given volume and surface forces� � � ��i� is the unit outward normal to

��� �e and �p are the elastic and plastic parts of the strain tensor and K�� �denote positive constants characterizing the elastic properties of the deformedbody� The symbol �D stands for the deviator of the tensor �� i�e�

�D � � � �

ntr ���

�� FUCHS et al�

where � is the identity and tr � � �ii�

Finally� the convex function F � M n�nsym R determines the yield surface in

the space of stresses� i�e� the set of tensors � s�t�

F��� � �������

Equation ����� is called the yield condition� since a necessary condition forj�p�x�j � � is exactly the equality F���x�� � �� We con�ne ourself to the vonMises yield condition by setting

F��� � j�Dj �p�k�� � � M n�n

sym �����

where k� is a positive constant� The plasticity model described by the relations����� � ����� and ���� is called Hencky�Il�yushin plasticity� Concerning themathematical treatment of other plasticity models we refer to the papers �An���RS�� �Se��� �Fu� and the references quoted therein�

The �rst serious problem we are faced with when studying the boundary valueproblem ����� � ������ ���� is connected with the correct functional formu�lation� Indeed� the primary version of a functional formulation based on theassumption of summability of the strain tensor has the form of the minimaxproblem� We seek a pair �u� �� � �V� � u���K which is a saddle point of theLagrangian �v� �� �v� �� on the set �V� � u���K� i�e�

�u� �� � �u� �� � �v� �� �v � V� � u�� � � K�����

where

�v� �� �Z�

���v� � � � g����� f � v� dx�Z���

F � v d�

V� � u� � fv � D������ � v � � on ���g � u���� �

denotes the set of kinematically admissible displacements�

K �n� � � � F���x�� � � for a�a� x � �

o

Variational Methods ��

is the set of admissible stress tensors� and we further let

Mn�nsym � � g���� �

�n�K�

tr�� � g���j�Dj��

R � t g���t� �

������t�� jtj � p

�k��

��� jtj p�k��

Dp�q��� �nv � L����Rn� � kvkp�q � kdiv vkLp����

�kjvj� j�D�v�jkLq��� � ��o� p� q � �� and

� �n� � ��ij� � k�k� � ktr �kL���� � k�DkL���� � ��

odenote the spaces of admissible �elds of displacements and stresses� respec�tively� Lp���Rm�� W �

p���Rm� are the familiar Lebesgue and Sobolev spaces of

vector�valued functions taking their values in Rm �compare �A� and �LU��� Itis known �see �MM��� �MM��� that the space D������ is imbedded continuous�ly in the spaces L

nn�� ���Rn� and L�����Rn� and compactly in the spaces

Lp���Rn� for p � ��� nn�� �� So the minimax problem ���� will be well�posed

at least formally if we assume that

f � Ln���Rn�� F � L������Rn�� u� � W �

� ���Rn�������

Unfortunately� the minimax problem ���� even for smooth data and undersome additional natural conditions which will be imposed below� has in ge�neral no solution for the following reasons� The Lagrangian generates twovariational problems being in duality� In the �rst of them a displacement �eldu � V� � u� is determined as a minimizer of the problem

I�u� � inffI�v� � v � V� � u�g������

where

�������V� � u� � v I�v� � supf�v� �� � � � Kg � R

�g���v��dx�M�v��

M�v� �R�f � v dx �

R���

F � v d�������

�� FUCHS et al�

Here g � M n�nsym R is equal to the conjugate function of g�� i�e�

g��� � supf� � � � g���� � � � M n�nsym g�

which can be expressed through the conjugate function of g�� in the followingway�

Mn�nsym � � g��� �

�K� tr

�� � g��j�Dj��

R � t g��t� � supfst� g���s� � s � Rg�

In the case of Hencky�Il�yushin plasticity we have

g��t� �

������t�� jtj � t� �

k�p��

k��p�jtj � k�

���� jtj t��

������

In the second variational problem� called the Haar�Karman principle� we lookfor a stress tensor � � Qf �K such that

R��� � supfR��� � � � Qf �Kg�������

where

Qf �K � � R��� � �u�� ��������

is the functional of the problem and

Qf � f� � � �Z�

� � ��v�dx � M�v� �v � V�g

is the set of stress tensors satisfying the equilibrium equations ����� in theweak sense�

As it will be shown below problem ������ has a unique solution � under thecondition K � Qf � �� Moreover� if the so�called �safe load hypothesis� isimposed� i�e�

��� � Qf and ��� � � F���� � ��� a�e� in ��������

Variational Methods �

then we have two important results�

inffI�v� � v � V� � u�g � R���������

�������������������

�u� �� � �V� � u���Kis a saddle point of problem ���� if and only if

u � V� � u� is a minimizer of problem ����� and

� � Qf �K is a maximizer of problem �������

�����

So� according to ������ the minimax problem ���� is solvable if and only ifthe variational problem ����� admits a solution� At this stage we should adda remark justifying the considerations from above� Suppose that u and � aresu�ciently regular solutions of ����������� with F de�ned in ����� Then itis easy to see that u solves ����� whereas � is R�maximizing� Conversely�any smooth saddle point �u� �� of the Lagrangian provides solutions of therelations ������������ From this point of view and also with respect to �����it is natural to investigate the variational problems ����� and ������ and toprove some smoothness properties of suitable generalized solutions� However�although the convex functional I is continuous on D������ and coercive onV� � u�� i�e�

I�v�� if kvk��� � and v � V� � u��

we can not prove the existence of a minimizer of problem ����� by applying thedirect method of the Calculus of Variations� The point is that the nonre�exivespace D������ shows the same �unpleasant� properties as the classes L� andW �

� �

In order to demonstrate that problems ���� and ����� in general have nosolutions� let us consider as an example the plane deformation of a concentricring whose interior contour is �xed whereas the exterior one is twisted� Let usintroduce polar coordinates � � �� with origin at the center of the ring and letR� and R� be the interior and exterior radius of the ring� Next� let u�� u� and���� ���� ��� be the physical components of the displacement �eld u and thestress tensor �� Our boundary conditions take the form

�����u � �u�� u��� ��� � �� u � � at � R��

u� � �� u� � U� � constant � at � R������ �

� FUCHS et al�

We suppose that

f � �� U� U� ��� � �

��

k���

R��������

where � � R��R��

Let us introduce the tensor � by setting

� �

�B� ��� ���

��� ���

�CA � ��� � ��� � �� ��� � k��R�

���������

It is easy to check that � � K and div � � � on � and thus

� � Qf �K�������

We let

u� � �u��� u���� u�� � �� u�� �

U�

U�

k���

� � R��

��������

If we introduce the function

u � � u�� u��� u� � �� u� �

R�U� �

U��� � �

R�� R�

��������

then elementary calculations show that

�����������������������

������ � ����u�� � ���� u��

����u�� � ����u�� � ���� u� � ���� u� � ��

u��R�� �U��U�

�� u���R�� � ��

u��R�� � u���R�� � U��

������

Next� let � be an arbitrary function from �� �K where

�s � f� � � � div � � Ls���R��g� s � ��������

Variational Methods �

By Lemma A���� � � L���� M ���sym� and therefore� according to Lemma A�����

a sequence �m � C���� M ���sym� exists such that

����������m � in L���� M ���

sym��

div �m div � in L����R���

k�mDkL���� � k�DkL�����

�����

From the last identity in ����� we infer that

�m � �� �K������

Now� let us consider the integral

Im �Z�

���u�� � ��m � �� � �u� � u� � div��m � ��� a��� �m � ��� dx

where

a��� �� ��

�K�tr� tr� �

���D � �D� �� � � M ���

sym ����� �

Since the integral contains only smooth functions we can integrate by parts�As a result we get

Im �Z��

��m � ��� � �u� � u� d�Z�

h�� u� � ��m � ��� a��� �m � ��

idx�

It follows from ������ that

�� u� � ��m � ��� a��� �m � �� � ������ u�� �

��������

m�� � ���� � ��

and therefore

Im �Z

��R�

��m � ��� � �u� � u�d � ���Z�

��m�� � �����u�� � u��

�����R�

R�d�

� ���Z�

��m���R��� k��U� � U�

�R�d��

�� FUCHS et al�

For all and � we have �by ������ j�m�� j � k� which gives Im � � for anym � N � After passing to the limit m� we get the variational inequality

Z�

���u�� � �� � �� � �u� � u� � div�� � ��� a��� � � ��� dx � �������

for all � � �� �K� For � � Qf �K inequality ������ reduces to

Z�

���u�� � �� � ��� a��� � � ��� dx � �

which means that � is the unique solution of the dual problem ������� Now� letus suppose that a pair �u� �� � �V� � u���K is a saddle point of the minimaxproblem ����� By ����� the stress tensor � is determined according to �������Since j�Dj � p

�k� inside of �� the plastic part of the deformation is equal tozero and we have

��u� ��

�K��tr � �

���D � �� u� in ��

Moreover� u � u at � R� implies u � u in �� Recall that u � ��� U��U��

� at � R�� On the other hand� u � V� � u� and therefore u � � at � R� whichleads to a contradiction�

Thus the assumption of summability of the strain tensor is too strong anddoes not enable us to prove existence theorems for problem ����� Hence weneed a suitable relaxed version of ���� whose dual problem is still given by������ and which will allow us to construct physically reasonable generalizedsolutions� A detailed analysis of this construction will be presented in the nextsections� It is worth to remark that in the present case minimizing sequencesof problem ����� converge to a function having a jump discontinuity on theinternal contour � � R�� which is in accordance with the theory of perfectelastoplasticity� Moreover� this theory admits discontinuities of slip type whichmeans that the normal component of the displacement �eld w�r�t� to a surfaceof discontinuity remains continuous�

Up to now we limited our discussion to the Hencky�Il�yushin model� At theend of chapter � the reader will �nd some comments concerning other modelstreated in plasticity with hardening� We also refer to chapter ��

Variational Methods ��

��� Relaxation of convex variational problems

in non re�exive spaces� General construc�

tion

Let V� U and P be Banach spaces so that V � U � and let V� be a subspace ofV � Next� let A � V P denote a linear bounded operator� and suppose thatG � P R and M � U R are convex� lower semicontinuous functionals �forgeneral terminology we refer the reader to the book �ET��� We also assumethat these functionals are proper in the sense that they are not equal to ��identically and do not take the value ���

We denote by P � and U� the dual spaces to P and U � by h�� �i and ��� �� theduality relations between P and P �� U and U�� respectively� and by G� theconjugate functional of G� i�e�

G��p�� � sup fhp�� pi �G�p� � p � Pg� p� � P ��

Let us consider the variational problem

to �nd u � V� � u� such that I�u� � inf fI�v� � v � V� � u�g�����

where

I�v� � G�Av� �M�v�

and uo � V is �xed� To formulate the problem dual to ����� we �rst introducethe Lagrangian by letting

�v� q�� � hq�� Avi �G��q�� �M�v�� q� � P �� v � V� � u��

Then the dual problem reads

to �nd p� � P � such that R�p�� � sup fR�q�� � q� � P �g�����

where

R�q�� � inf f�v� q�� � v � V� � u�g�

Concerning the solvability of problem ����� we have the following

�� FUCHS et al�

THEOREM ����� ��ET��� Suppose that the next two conditions hold

C �� inf fI�v� � v � V� � u�g ��������������

������u� � V� � u� � G�Au�� � ���M�u�� � ���

the function p G�Au� � p� is continuous at zero������

Then problem ����� has at least one solution and the identity

C � sup fR�q�� � q� � P �g����

is valid�

Proof� We consider the variational problem equivalent to �����

�����������������������

to �nd u � V such that I�u� � inff I�v� � v � V g where

I�v� � G�Av� � M�v��

M�v� �

��� M�v�� v � V� � u�

��� v � V� � u�� v � V�

����

Now� let us de�ne the perturbed functional

!�v� p� � G�Av � p� � M�v�� v � V� p � P�

Following �ET� we introduce the problem dual to ����� i�e�

to �nd p� � P � such that � !���� p�� � sup f � !���� q�� � q� � P �g���� �

Here !� � V � � P � R is the conjugate function of !� In our case we have

�!���� q�� � � supv�V

supp�P

fhq�� pi �G�Av � p�� M�v�g

� � supv�V

supq�P

f � hq�� Avi� hq�� qi �G�q�� M�v�g

� � supv�V

f � hq�� Avi�G��q��� M�v�g � infv�V�u�

�v� q�� � R�q���

Variational Methods ��

Thus problem ��� � is equivalent to problem ������ Su�cient conditions underwhich problem ��� � is solvable and we have the identity

inf f I�v� � v � V g � supf�!���� q�� � q� � P �g�����

are for example �see �ET�� Chapter III� Proposition ����

inf f I�v� � v � V g ��������������

the function ! is convex�������

������ u� � V such that the function p !� u�� p�

is �nite and continuous at zero�������

Obviously ����� is equivalent to condition ������ Convexity of ! is providedby convexity of G and M � Finally� condition ������ follows from ����� if weput u� � u�� Since identities ����� and ���� are equivalent� Theorem ����� isproved�

Along with the problems ����� and ����� we consider the following minimaxproblem

�����to �nd a pair �u� p�� � �V� � u��� P � such that

�u� q�� � �u� p�� � �v� p��� �v � V� � u�� q� � P ��������

�Such a pair is called a saddle point��

Since G � P R is a proper� convex� lower semicontinuous functional� we have

G�p� � sup fhp�� pi �G��p�� � q� � P �g������

and therefore

I�v� � sup f�v� q�� � q� � P �g�������

�� FUCHS et al�

Thus under conditions ������ ����� we have the identity

infv�V�u�

supq��P �

�v� q�� � C � supq��P �

infv�V�u�

�v� q�������

and the general theory of duality provides the following statement������������������������

a pair �u� p�� � �V� � u��� P �

is saddle point of the minimax problem ������if and only if

u � V� � u� is a minimizer of problem ����� and

p� � P � is a maximizer of problem ������

�����

So� by Theorem ����� and ������ solvability of problem ����� is equivalent tosolvability of the minimax problem �������

From now on we will impose the following additional conditions on the data������������the embedding of V into U is continuous�

V� is dense in U �

U is a re�exive space�

���� �

������u� � V� � u� � u� � int dom M�

dom M �� fu � U � M�u� ��g�������

I�v� �� if kvkV �� and v � V� � u��������

Condition ������ means coercivity of I on the set V� � u��

Standard arguments now show that if the space V is re�exive� then the coerci�vity condition ������ together with convexity and lower semicontinuity of thefunctionals G andM provide existence of at least one solution of problem �����and thus existence of at least one solution of the minimax problem ������� Soit remains to discuss the case of a nonre�exive space V �

The example considered in the previous section shows that if the space V isnonre�exive� then in general problems ����� and ������ have no solutions� Wetherefore have to construct suitable relaxations which should satisfy at leasttwo restrictions�

Variational Methods ��

�� conservation of the greatest lower bound of problem ������

�� conservation of the dual problem�

The �rst condition is clearly satis�ed if we can guarantee that all minimizingsequences of problem ����� converge in some sense to a solution of the relaxedproblem� As to the second restriction one should say that a solution of thedual variational problem exists and as a rule� it is unique and it has a clearphysical or geometrical meaning� Thus there is no need to change the dualvariational problem�

The construction of a suitable variational relaxation to problem ����� nowfollows the paper �Se��� We �rst introduce an auxiliary operator A� whosedomain D�A�� is the set

P �� � fp� � P � � �u� � U� such that hp�� Aui � �u�� u� �u � V�g�

By condition ���� �� for each p� � P �� � there is only one element u� � U�

satisfying the identity hp�� Aui � �u�� u� on V�� So we can de�ne the linearoperator A� � P �

� U� just through the relation

hp�� Aui � �A�p�� u� �p� � P �� � u � V��

If u� is a �xed element from V � then we have the identity

hp�� Aui � E�u�� p�� � �A�p�� u� �u � V� � u�� p� � P �

�������

where

E�u�� p�� � hp�� Au�i � �A�p�� u���

We enlarge the set V� � u� by letting

V � fu � U � supkp�kP����p��D�A��

jE�u�� p�� � �A�p�� u�j � ��g�������

and introduce a relaxation ! of the functional I with the help of the LagrangianL through the relations�����������������

L�v� q�� � E�u�� q�� � �A�q�� v��G��q�� �M�v��

q� � D�A��� v � V�

!�v� � supq��D�A��

L�v� q��� ! � V R �

������

�� FUCHS et al�

Let us remark some useful facts which follow from ������ and our de�nitions������ and �������

V� � u� � V�������

!�v� � I�v� �v � V� � u��������

Indeed� if v � V� � u�� then we have from ������

supkp�kP����p��D�A��

jE�u�� p�� � �A�p�� v�j �

� supkp�kP����p��D�A��

jhp�� Avij �

� supkp�kP���

jhp�� Avij � kAvkP �

and inclusion ������ follows� Next� by ������

L�v� q�� � �v� q�� �v � V� � u�� q� � D�A��������

But then� for v � V� � u�� we have �recall �������

!�v� � supq��D�A��

�v� q�� � supq��P �

�v� q�� � I�v��

LEMMA ����� Suppose that for any p � domG� there exists a sequencep�m � D�A�� such that

�����p�m

�� p�in P ��

G��p�m� G��p��������

Then the identity

!�v� � I�v� �v � V� � u����� �

is valid�

Variational Methods �

Proof� Since G is proper� convex and lower semicontinuous� the functionG� is of the same type� Thus

G�p� � supfhq�� pi �G��q�� � q� � dom G�g

and therefore

I�v� � supf�v� q�� � q� � dom G�g �v � V� � u��������

Suppose �rst that I�v� � ��� Then� for any A �� there is a q� � dom G�

such that �v� q�� A� Using the conditions of Lemma ����� one can �nd asequence p�m � D�A�� having the properties

p�m�� q� in P �������

and

G��p�m� G��q��� �v� p�m� �v� q���������

It follows from the de�nition of ! and identity ����� that

!�v� � L�v� p�m� � �v� p�m� �v� q�� A�

By the arbitrariness of A� we get !�v� � ���

If I�v� � ��� then according to ������ for any � � there exists q� � dom G�

such that

I�v� � �v� q�� � ��

Let a sequence p�m � D�A�� be chosen such that ������ and ������ hold� Then

I�v� � �v� q�� � � � limm�� �v� p�m� � � � lim

m��L�v� p�m� � � � !�v� � ��

From this we get !�v� � I�v� and Lemma ����� is proved��

It is interesting to know under what conditions the following statement will betrue

V � V� � u��������

� FUCHS et al�

LEMMA ����� Suppose that

P is a re�exive space�������

the restriction of the operator A on V� has a closed rangeof values in P�

������

kwk � supp��D�A���kp�kP���

j�A�p�� w�j � �� w � ��������

Then ������ holds�

Proof� We denote by A the restriction of the operator A on V�� Let us takean arbitrary element u � V and set w � u � u�� For the linear functionalp� �A�p�� w� we have the estimate

j�A�p�� w�j � kwk kp�kP � �p� � D�A���

Since u � V we have kwk � ��� By condition ������ and the Hahn�Banachtheorem we �nd an element p � P such that

hp�� pi � �A�p�� w� �p� � D�A��������

Let us introduce the set

Q � fp � P � hp�� pi � � for any p� � ker A�g�

It is clear that A�V�� � Q� Suppose that there is some p� � Q n A�V��� Bycondition ������ the set A�V�� is closed� hence there exists p�� � P � such that

hp��� p�i � �� hp��� pi � � �p � A�V���

Thus hp��� Avi � � for all v � V� which shows p�� � D�A�� and �A�p��� v� � �for any v � V�� By condition ���� � p�� � ker A�� But then it follows from thede�nition of the set Q that hp��� p�i � �� So we have proved that Q � A�V���This means that for p satisfying identity ������ we can �nd v � V� with theproperty p � Av and �A�p�� w � v� � � for all p� � D�A��� From this and������ it follows that w � v� hence u � V� � u� and Lemma ����� is proved�

Variational Methods �

REMARK ����� We have shown that if conditions ���� �� ������ ������������� hold� then

V � V� � u� and !�v� � I�v� �v � V������

Before proving the main results concerning the relaxation of problems �����and ������ we establish

LEMMA ����� Consider a bounded sequence fumg � V� � u� converging tou weakly in U � Then

u � V����� �

lim infm�� I�um� � !�u��������

Proof� We have

jE�u�� p�� � �A�p�� um�j � jhp�� Aumij � kAk supmkumkV

for all p� � D�A�� such that kp�kP � � �� Passing to the limit and using thede�nition of the class V we get ���� �� To prove ������ we take into accountidentity ����� in order to get

I�um� � �um� q�� � L�um� q

�� �

E�u�� q�� � �A�q�� um��G��q�� �M�um��

q� � D�A���

and therefore

lim infm�� I�um� � E�u�� q�� � �A�q�� u��G��q��� i�e�

lim infm�� I�um� � L�u� q�� �q� � D�A���

From this and the de�nition of the functional ! �see ������� ������ follows�Lemma ����� is proved�

�� FUCHS et al�

We consider now the minimax problem

�����to �nd a pair �u� p�� � V �D�A�� such that

L�u� q�� � L�u� p�� � L�v� p�� �v � V� q� � D�A���������

As usual this problem generates two variational problems being in duality�

�����to �nd u � V such that

!�u� � inf f!�v� � v � Vg�������

where

!�v� � sup fL�v� q�� � q� � D�A��g�

and

�����to �nd p� � D�A�� such that

R�p�� � sup f R�q�� � q� � D�A��g�������

where

R�q�� � inf fL�v� q�� � v � Vg�

The main statement of this section is contained in

MAIN THEOREM ����� Suppose that the conditions ������ ������ ���� �������� hold� Then we have

�i� Problems ������ and ������ are solvable� Moreover� if u � V is a solutionto problem ������ and p� � D�A�� is a solution to problem ������� thenthe identity

!�u� � C � R�p��������

is true�

Variational Methods ��

�ii� Problems ����� and ������ are equivalent� i�e� they have the same set ofsolutions�

�iii� A pair �u� p�� � V � D�A�� is a saddle point of the minimax problem������ if and only if u � V is a minimizer of problem ������ and p� inD�A�� is a maximizer of problem �������

�iv� Any minimizing sequence of problem ����� contains a subsequence con�verging to some solution to problem ������ weakly in U �

Proof�

�i� Let fumg � V� � u� be an arbitrary minimizing sequence of problem������ i�e� I�um� C � inffI�v� � v � V� � u�g� Condition ������implies boundness of this sequence in V and� by ���� �� it contains asubsequence converging weakly in U � We denote by u its limit� Due toLemma ����� we have u � V and !�u� � C� Let p� be any solution toproblem ������ If we can show that u is a solution to problem ������� p�

is a solution to problem ������ and identity ������ holds� then statement�i� will be established� Moreover� by the arbitrariness of the minimizingsequence� statement �iv� and also the fact that any maximizer of problem����� is a maximizer of problem ������ will be proved�

According to ���� we have

!�u� � C � R�p��������

and therefore

L�u� q�� � C � �v� p�� �q� � D�A��� v � V� � u��������

Let us show that p� � D�A��� To do this we rewrite the right hand sideof inequality ������ in the form

�hp�� Avi � �C � hp�� Au�i �G��p�� �M�u� � v� �v � V��

By condition ������ the functional M is continuous at the point u� andtherefore it is bounded in some ball of the space U with center at u�� Sowe have the estimate

�hp�� Avi � constant kvkU �

�� FUCHS et al�

It follows from the Hahn�Banach theorem that there exists u� � U� suchthat

hp�� Avi � �u�� v� for all v � V��

This gives the claim p� � D�A��� Using identity ������ we may rewriteinequality ������ in the equivalent form

L�u� q�� � C � L�v� p�� �q � D�A��� v � V� � u�������

Now we are going to prove that the right hand side of the last inequa�lity holds for all v � V� Obviously� it is enough to check this for allv � V � dom M �

First let w � V � int dom M � Since w � u� � U � condition ���� � givesexistence of a sequence fvmg � V� such that vm w � u� in U � Thefunctional M is continuous on int dom M � hence M�vm � u��M�w��Since vm � u� � V� � u� for all m � N� we can insert vm � u� into �����and get after passing to the limit ����� for any v � V � int domM �In case w � V � dom M we set v��� � �� � ��w � �u� � int dom Mfor all � ���� ��� For v � v��� inequality ����� is correct� The functionv L�v� p�� is convex and thus

C � ��� ��L�w� p�� � �L�u�� p���

With � going to zero we conclude that�����u � V� p� � D�A��

L�u� q�� � C � L�v� p�� �v � V� q� � D�A��������

From ����� we get

infv�V�

supq��D�A��

L�v� q�� � infv�V�

!�v� � !�u� �

� C � R�p�� � supq��D�A��

R�q�� � supq��D�A��

infv�V�

L�v� q��

and therefore

!�u� � infv�V�

!�v� � C � R�p�� � supq��D�A��

R�q������� �

So we have shown that u is a minimizer of ������� p� is a maximizer of������ and identity ������ holds�

Variational Methods ��

�ii� To prove statement �ii� it is enough to check that any maximizer p�� inD�A�� of problem ������ is a maximizer of problem ������ By ���� ������� and ����� we have

C � R�p��� � infv�V�

L�v� p��� � infv�V�u�

L�v� p���

� infv�V�u�

�v� p��� � R�p��� � supq��P �

R�q�� � C

and the claim follows�

�iii� This statement is a consequence of identity ������ and standard argu�ments from duality theory�

�iv� This statement has already been proved �see �i��� Theorem ����� is esta�blished�

Taking into account the statements of Theorem ����� it is quite natural to callall solutions to problem ������ weak solutions to the variational problem ������and all solutions to problem ������ weak solutions to the minimax problem�������

�� FUCHS et al�

��� Weak solutions to variational problems of

perfect elastoplasticity

We consider a model of a perfect elastoplastic body providing some generaliza�tion of the classical Hencky�Il�yushin plasticity model� Suppose that an evenconvex function g� � R R of class C� is given satisfying the conditions

�����g���� � �� g����� � �� and

g���t�p�k� as t ���

�����

where k� is a �xed positive constant� In the case of Hencky�Il�yushin plasticityg� has the form �see �������

g��t� �

����t� if jtj � t� �

k�p��

k��p�jtj � k�

��� if jtj t��

where the positive constant � is related to the elastic properties of the bodyunder consideration� Another example of a function g� satisfying ����� is givenby

g��t� �p�k��

p� � t� � ��������

It is easy to see that for the conjugate function g�� of any g� with property����� we have

g���s� � supfst� g��t� � t � Rg � �� if jsj p�k�������

Setting

g��� ��

�K�tr

�� � g��j�Dj�� � � M n�n

sym ������

K� denoting a positive constant� we get

g���� � supf� � � � g��� � � � M n�nsym g �

��n�K�

tr�� � g���j�Dj� �� � M n�nsym �

����

Variational Methods ��

Now we may consider the minimax problem ���� for the Lagrangian

�����������������

�v� �� �R����v� � � � g�����dx�M�v��

v � V� � u�� � � K�

where M�v� �R�f � vdx�

R���

F � vd�

So we look for a pair �u� �� � �V� � u���K such that

�u� �� � �u� �� � �v� �� �v � V� � u�� � � K�����

We recall the de�nitions of the sets V� and K which were given in the �rstsection�

V� � fv � D������ � v � � on ���g�

K � f� � � � F���x�� � j�D�x�j �p�k� � � for a�a�x � �g�

where

Dp�q��� � fv � L����Rn� � kvkp�q � kdiv vkLp��� � kjvj� j�D�v�jkLq��� � ��g�p� q � ��

� � f� � ��ij� � k�k� � ktr�kL���� � k�DkL���� � ��g�

For the space D������ the following facts are known �see� for example� �MM����MM����

�����������the space D������ is embedded continuously into the spaces

Ln

n�� ���Rn� and L�����Rn�� and compactly into the spaces

Lp���Rn� for any p � ��� nn�� ��

��� �

It is therefore quite natural to impose the following conditions on the data ofthe problem�

f � Ln���Rn�� F � L������Rn�� u� � D������������

�� FUCHS et al�

As usual the minimax problem generates two variational problems being induality�

to �nd u � V� � u� such that I�u� � inffI�v� � v � V� � u�g������

where

I�v� � sup��K

�v� �� �Z�

g���v�� dx�M�v�� v � D�������������

and

to �nd � � Qf �K such that R��� � supfR��� � � � Qf �Kg�������

where

R��� �

��� �u�� �� if � � Qf �K�� if � � Qf �K

� � � � K�������

and the set Qf is de�ned as

Qf � f� � � �Z�

� � ��v� dx � M�v� �v � V�g�������

In order to handle the variational problem ����� with the help of the generalscheme described in the previous section� we set

V � D������� U � Ln

n�� ���Rn�� U� � Ln���Rn��

�u�� u� �R�u� � u dx�

�����P �

np � f�� ag � kpk�P � k�Dk�L���� � �

nktr�k�L�����

�kak�L������ � ��o� L���� M n�n

sym �� L������Rn��

������

It is clear that�����P � �

np� � f�� bg � �D � L���� M n�n

sym �� tr� � L�����

b � L������Rn�o� �� L������R

n������

Variational Methods �

and also that�p � f�� ag � P� p� � f�� bg � P �

hp�� pi �Z�

� � � dx �Z���

a � b d�

The norm on the space P � is de�ned in the usual way

kp�kP � � supfhp�� pi � kpkP � �g�

This norm is equivalent to the following one

�k�Dk�L���� �

nktr�k�L���� � kbk�L������

����

Obviously the spaces V � U � V� satisfy the conditions stated in ���� ��

Next� let us introduce the functionals G � P R and M � U R

�������G�p� �

R�g���dx�

R���

F � a d� p � f�� ag � P�

M�v� � � R�f � v dx� v � U�

�����

These functionals are convex and �nite and therefore continuous� It is also clearthat condition ������ imposed on the functional M is satis�ed� Moreover� wehave

G��p�� � supfhp�� pi �G�p� � p � Pg �

���R�g����dx if b � F on ���

�� if b � F on ���

� � p� � f�� bg����� �

The linear operator A � V P may be introduced as follows� We let

Av � f��v���v������g� v � V�

and get from ��� �

kAvkP ���nkdiv vk�L���� � k�D�v�k�L���� � kvk�L������

��� � c��� n�kvk����

� FUCHS et al�

hence the operatorA is bounded� Taking into account all de�nitions introducedabove� we may represent our functional I � V R in the form

I�v� �Z�

g���v��dx�Z���

F � v d�Z�

f � v dx � G�Av� �M�v��

Now we check if conditions ������ ����� and ������ hold� Since the functionalG is convex and �nite� i�e� dom G � P � the function p G�Au� � p� iscontinuous at zero for any u� � V� � u�� By �niteness of the functional Mcondition ����� is ful�lled�

The validity of conditions ����� and ������ is guaranteed by the so called �safeload condition�� i�e� we assume that

��� � Qf � F���� � �p�k���� �� a�e� in � for some � ���� ���������

Indeed� by convexity of the function g� and by condition ����� we may �nd anumber s � such that g���s� �

p�k� �� � Then g��t� � g��s��

p�k� �� �t�

s� for all t and therefore

I�v� �

��K�

Z�

div�v dx �Z�

hg��s� �

p�k�

� � �

��j�D�v�j � s�

idx�M�v�

� �g��s��p�k� �� s�j�j� �

�K�

Z�

div�v dx

�p�k� ��

Z�

j�D�v�jdx�M�v�� �Z�

�� � ��u��dx�Z�

�� � ��v�dx

� ��K�

Z�

div�v dx �p�k�

�� �

Z�

j�D�v�jdx�

� �n

Z�

jtr��j jdivvjdx�Z�

�� � ��u��dx�M�u���

��g��s��p�k� �� s�j�j �� as kvk��� ��� v � V� � u��

������

So ������ follows from ������� Finally� condition ����� is provided by the esti�mate

C � inffI�v� � v � V� � u�g � R���� ���

Variational Methods �

There is a useful criterion which implies condition ������� Let us set

�� � inffp�k�

Z�

j��v�jdx � v � V�� div v � � in �� M�v� � �g�

As it was shown in �MM�� there exists a stress tensor �� � K such thatZ�

�� � ��v�dx � ��M�v� �v � V��

Suppose that the loads f and F and the domain � are such that

�� ��������

Then� for � � �� and �� � ���� we get �� � Qf and j��Dj � �j�D� j � �

p�k��

i�e� condition ������ implies �������

To sum up� we see that conditions ����� and ������ guarantee the validity ofconditions ������ ������ ���� ��������� and we can state that problem ������ hasat least one solution � � Qf �K� that identity ����� holds and that statement����� is valid� As it was demonstrated in the �rst section the variationalproblem ����� and the corresponding minimax problem ���� in general haveno solutions� Their relaxations� providing existence of weak solutions� may nowbe constructed with the help of the abstract scheme of relaxation for convexvariational problems in nonre�exive spaces which was discussed in the secondsection� To do this we need to de�ne the auxiliary operator A� � D�A�� U��A pair p� � f�� bg belongs to D�A�� if and only if there is an element u� �U� � Ln���Rn� such thatZ

u� � v dx �Z�

� � ��v�dx�Z���

b � v d �v � V��������

From ������ it follows that if p� � f�� bg � D�A��� then u� � A�p� � �div � �Ln���Rn�� So we have

D�A�� � fp� � f�� bg � P � � div � � Ln���Rn��

Z���

b � v d �Z�

�� � ��v� � v � div ��dx �v � V�g�

In the �rst section �see ������� we introduced the space

�s � f� � � � div � � Ls���Rn�g� s � ��

�� FUCHS et al�

DEFINITION ����� Let � be the unit outward normal to the boundary ���Let � � �n and b � L������R

n�� We say that �� � b on ��� if and only iff�� bg � D�A���

This de�nition of course is in accordance with the usual pointwise identity�� � b on ��� provided the functions � and b are smooth enough� So

�� � b on ���� � � �n� b � L�����Rn�� p� � f�� bg � D�A��������

is a suitable extension of the classical notion�

Now we can describe the extension V of the set V� � u�

�����������V �

nv � L

nn�� ���Rn� � sup

kp�kP����p��f��bg�D�A��D�Z���

b � u� d�Z�

�� � ��u�� � �u� � v� � div �� dxE� ��

o�

������

From the de�nition of the class V it follows directly that V � BD���Rn�where BD���Rn� is the space of functions of bounded deformation �see �MSC���Su�� �ST���� Functions from this set are summable and the symmetric part oftheir gradient is a bounded measure� The norm in BD���Rn� can be introdu�ced for example as follows

kvkBD��Rn� �Z�

jvjdx�Z�

j��v�j�

where

Z�

j��v�j � sup

����Z�

v � div � dx � � � C�� ��� M n�n

sym �� j� j � � in �

� �

By conditions ������ ���� � G��q�� � G��f�� bg� � �� if b � F on ��� or� � K� For this reason we consider the relaxed Lagrangian of the form

�����������L�v� q�� � E�u�� q�� � �A�q�� v��G��q�� �M�v�

� �Z���

F � u�d�Z�

h��u�� � � � �u� � v� � div � � g����� f � v

idx

������

Variational Methods ��

for all v � V and q� � f�� Fg � D�A�� such that � � K� It is convenient tointroduce the set

Q � f� � �n � �� � F on ���g � f� � � � f�� Fg � D�A��g�����

and the new Lagrangian on V � �Q �K�

L�v� �� � L�v� q�������

where

q� � f�� Fg � D�A��� � � K�

Now� instead of the minimax problem ����� we consider its relaxation

�����to �nd a pair �u� �� � V � �Q �K� such that

L�u� �� � L�u� �� � L�v� �� �v � V� � � Q �K����� �

For the functional ! � V R we have the formula

!�v� � supq��D�A��

L�v� q�� � supq��f��Fg�D�A��

��K

L�v� q�� � sup��Q�K

L�v� ���������

and the relaxation of the variational problem ����� reads

to �nd u � V such that !�u� � infv�V�

!�v��������

We next give a su�cient condition which implies that ! � I on the set V��u��

LEMMA ����� Suppose that

�Z�

�p�k� � g���t�� dt � ���������

Then we have

!�v� � I�v� �v � V� � u��������

�� FUCHS et al�

Proof� Recall that

g���p�k�� � supf

p�k�t� g��t� � t � Rg�

By convexity of g� and condition ����� we have

g���p�k�� � lim

t��

tZ�

�p�k� � g������d� � g����

and ������ implies

jg���p�k��j � ���������

Since the function g�� is convex and lower semicontinuous� ������ gives

g�� � C����p�k��

p�k��

�������

so that

�c � � jg���t�j � c �t � ��p�k��

p�k���������

Obviously in our case

p� � f�� bg � dom G� � � � K� b � F on ����

By Lemma A���� �see Appendix� for t � � a sequence �m � C���� M n�nsym �

exists such that���������������

�m � in L���� M n�nsym �

�mD �� �D in L���� M n�n

sym �

k �mDkL���� � k�DkL������ �m � K��

�����

Let us take a sequence �m � C�� ��� having the properties

� � �m � � on �� �m � a�e� in �

and let �m � �m �m � �� � ���� where �� is the function from the �safeload condition� ������� It is clear that �m � K� On the other hand� since

Variational Methods ��

div �� � �f a�e� in � and f � Ln���Rn�� we have �� � �n� But according toLemma A��� �see Appendix� �� � Ln��� M n�n

sym �� So we can deduce

�m � �n and div �m � ��� �m�div �� � �mdiv �m � � �m � ���r�m

for all m� Let us show that p�m � f�m� Fg � D�A��� i�e� �m � Q� In fact� bycondition ������� we have for all v � V�Z

�m � ��v�dx �Z�

��m �m � ��� �m��

�� � ��v�dx �

�Z�

�� � ��v�dx�Z�

�m� �m � ��� � ��v�dx �

�Z�

f � v dx�Z���

F � v d�Z�

�mdiv� �m � ��� � v dx

� R�� �m � ��� � �v �r�m�dx �

�Z���

F � v d�Z�

���� �m�div�� � �m div �m� � v dx

�Z�

�� � �v �r��� �m��dx�Z�

�m � �v �r�m�dx �

�Z���

F � v d�Z�

v � div �mdx�

Taking into account the estimate�Z�

j�m � �j�dx��� � �Z

��mj �m � �j�dx

����

�Z�

��� �m��j�� � �j�dx

����

we obtain�������������������������

�m � in L���� M n�nsym ��

�m � a�e� in ��

�mD �� �D in L���� M n�n

sym ��

�m � K� f�m� Fg � D�A���

�����

�� FUCHS et al�

Now from ������� ������� ����� and Lebesgue�s theorem on dominated conver�gence it follows that

G��p�m� �Z�

� �

�n�K�tr��m � g���j�mDj�

�dx

Z�

� �

�n�K�

tr�� � g���j�Dj��dx � G��p�� as m��

But since D�A�� � p�m �� p� � f�� Fg in P �� we get that all conditions of

Lemma ����� hold and therefore identity ������ is valid� Lemma ����� is pro�ved�

It remains to reformulate the Main Theorem ����� of the second section usingthe new de�nitions �see ������� ������ ������ ��������

THEOREM ����� Suppose that the conditions ������ ������ ������ hold� Thenthere exists at least one pair �u� �� � V � �Q � K� being a solution to theminimax problem ���� �� Moreover� � is a solution to the dual variationalproblem ������� u is a solution to the relaxed variational problem ������ andthe identity

!�u� � inffI�v� � v � V� � u�g � L�u� �� � R���

holds�

A pair �u� �� � V � �Q � K� is a saddle point of minimax problem ���� � ifand only if u � V is a minimizer of problem ������ and � � Qf � K is amaximizer of problem �������

For any v � V� � u�� we have

!�v� � I�v��

and if for example we assume that

�Z�

�p�k� � g���t��dt � ���

Variational Methods ��

then

!�v� � I�v� �v � V� � u��

Finally� any minimizing sequence of problem ����� converges strongly in L����Rn�and weakly in L

nn�� ���Rn� to some solution to problem �������

The statements of Theorem ����� allow us to call the pair �u� �� � V��Q�K�a weak solution to the minimax problem ����� and the function u � V a weaksolution to problem ������

Let us set

t� � supft � � g���t� �p�k�g����� �

Note that in the case of Hencky�Il�yushin plasticity t� �p�k���

� ��� whereas

t� � �� in the case of example ������

Suppose that

g� is of class C� on �� t�� t�� and g��� � on �� t�� t���������

LEMMA ����� Let the conditions ������ ������ ������ and ������ be satised�Then the variational problem ������ has a unique solution�

Proof� It is enough to prove that the function g�� is strictly convex� In fact�under condition ������� the function g�� is the Legendre transformation of g� inint dom g��� i�e�

t � g���s�� g���t� � sg���s�� g��s�� g

����� t�� t��� ���

p�k��

p�k���

Since �g������t� � ��g����s� for t � g���s� and any s �� � t�� t��� the function g�� is

strictly convex on ��p�k��p�k���

If g����p�k�� � ��� then g�� is strictly convex on ��p�k��

p�k��� Let

g���p�k�� � ��� We assume that for some t � ���

p�k�� there is a number

� ���� �� such that

�g���p�k�� � ��� ��g���t� � g���t��

�� FUCHS et al�

where t �p�k�� � �� � ��t� Then� by the Lagrange theorem� we can �nd

numbers t� and t�� such that

t � t� � t�� �p�k�� �g���

��t�� � �g�����t��� �

g���p�k��� g���t�p�k� � t

But this contradicts the monotonicity of �g���� on ���

p�k��� Since the function

g�� is even� we have shown that g�� is strictly convex on ��p�k��p�k��� Thus

the function g� de�ned by identity ���� is strictly convex� Lemma ����� isproved�

Let us return to the example considered in the �rst section� We observe �rstthat if ��� � �� then Q � �n� Thus the variational inequality ������ isequivalent to the following one

L� u� �� � L� u� �� � L�v� �� �� � Q �K� v � V�

with functions u and � being determined by the relations ������ and �������If we show that u � V� then since � � Q � K� the pair � u� �� will be aweak solution to the minimax problem ����� It is clear that in this case allconditions of Lemma ����� hold and therefore problem ������ has the uniquesolution de�ned by relations ������� We know that problem ���� � admits at

least one solution �u� �� � V � �Q �K�� where � is the function from �������

and that the inequality

eL�u� �� � L�u� �� �� � �n �K

holds which is equivalent to the relationZ�

h��u�� � �� � �� � �u��

u� � div�� � ��� a��� � � ��idx � �

�� � �n �K�

Since j�Dj � p�k� in �� we get

���������u� � �

�K��tr� � �

���D � ��eu� in ��

u� eu at � R��

Variational Methods �

This implies thatu� eu in �� But then eu � V and thus the pair �eu� �� is the

weak solution to the minimax problem ����� and this solution is known to beunique�

We �nish this chapter by remarking that for Hencky�Il�yushin plasticity thequestion concerning the relaxation of the variational problem ����� �in this casethe function g� is given by ������� was studied by G� Anzellotti and M� Gia�quinta �AG��� �AG��� R� Temam and G� Strang �ST��� R� Kohn and R� Temam�KT�� R� Hardt and D� Kinderlehrer �HK� and by R� Temam �T�� For the caseof bending perfect elastoplastic thin plates we refer the reader to �Se��� �CLT��

Chapter �

Di�erentiability properties of

weak solutions to boundary

value problems in the

deformation theory of plasticity

��� Preliminaries

In the �rst section of this chapter we brie�y recall some of the concepts whichhave been introduced in chapter �� in particular� we review how to formulatean appropriate relaxed version of the minimax problem describing the equi�librium con�guration of a perfect elastoplastic body� We then collect someresults concerning the structure of the corresponding weak solutions� The �rsttheorem states that the stress tensor is weakly di�erentiable with derivativesin the space L�

loc� For the Hencky�Il�yushin plasticity model we get� roughlyspeaking� that the elastic zone is an open set� As a byproduct we obtain apartial regularity theorem for strictly convex variational integrals with lineargrowth� The second section introduces some regularisations of our problem andwe prove convergence of these approximations to the original problem� Secti�on three is devoted to the proof of weak di�erentiablity of the stress tensor�moreover� we discuss some estimate of Caccioppoli type which is an essenti�al tool in our discussion of regularity� The fourth section contains a versionof a Campanato type estimate being valid for solutions to systems of partialdi�erential equations occuring in linear elasticity� In the next section we pro�ve certain decay estimates for quantities involving the mean oscillation of thestress tensor and also of the displacement �eld calculated with respect to balls

Variational Methods �

in the domain of de�nition� Here we argue by contradiction using a blow�upprocedure� In section six we apply the excess decay lemma by the way gettingthe regularity results stated in section one� A list of open problems concerningperfect plasticity is given in section seven� some comments on plasticity withpower hardening can be found in a �nal section�

�� FUCHS et al�

��� Formulation of the main results

For the reader�s convenience we recall the functional formulation of the classicalboundary value problem describing the equilibrium of a perfect elastoplasticbody within the framework of the deformation theory of plasticity and brie�ysketch the ideas leading to the notion of weak solutions� So �see Chapter ��section � for details� our problem is

�����to �nd a pair �u� �� � �V� � u���K such that

�u� �� � �u� �� � �v� �� �v � V� � u�� � � K������

where

�v� �� �Z�

���v� � � � g�����dx�M�v�

is the Lagrangian of the minimax problem ������ We let

V� � fv � V � v � � on ���g�

V � D������ �nv � L����Rn� � kvk��� � kdiv vkL�����

�kvkL���� � k�D�v�kL���� � ��o�

K �n� � � � F��� � j�Dj � p�k� � � a�e� in �

o�

� �n� � ��ij� � k�k� � ktr �kL���� � k�DkL���� � ��

o�

M�v� �R�f � v dx�

R���

F � v d�

g���� � ��n�K�

tr�� � g���j�Dj� � supn� � � � g��� � � � M

n�nsym

ois the conju�

gate function of g� M n�nsym R � g��� � �

�K�tr

�� � g��j�Dj�� � � M

n�nsym � and

g���s� � supfst� g��t� � t � Rg denotes the conjugate function of g� � R R �

Variational Methods ��

We suppose that the given functions f � F � u� and g� satisfy the followingconditions�

f � Ln���Rn�� F � L������Rn�� u� � V������

���������������

g� is an even convex function of class C��

g���� � �� g����� � ��

g���t� �p�k� as t ���

�����

In addition� it is assumed that � is a bounded Lipschitz domain in Rn whoseboundary �� consists of two measurable parts satisfying

��� � ��� � �� ��� � ��� � ���

The minimax problem ����� generates two variational problems being in dua�lity�

�����to �nd u � V� � u� such that

I�u� � inffI�v� � v � V� � u�g������

where

I�v� � sup��K

�v� �� �Z�

g���v��dx�M�v� �v � V�

and

�����to �nd � � Qf �K such that

R��� � supfR��� � � � Qf �Kg�����

where

R��� �

������u�� ��� � � Qf �K

��� � � Qf �K

��� � � � K�

� FUCHS et al�

Qf � f� � � �Z�

� � ��v�dx � M�v� �v � V�g�

We remark that according to De�nition ����� we have

� � Qf � div � � �f in �� �� � F on ����

i�e� the stress tensor � satis�es the equilibrium equations of stresses in suitableweak sense�

We also assume that the �safe load condition� holds� i�e�

�������� � Qf and � ���� �� such that

F���� � �p�k���� �� a�e� in ������

As it was shown in Chapter �� section �� ���� provides coercivity of the func�tional I on the set V� � u�� i�e�

I�v� �� if kvk��� �� and v � V� � u����� �

Under conditions ������ ������ ���� problem ���� has at least one solution whileproblems ����� and ����� in general are not solvable� In Chapter �� section �� wepresented suitable relaxations of problems ����� and ����� which have solutionsand these solutions were called weak solutions of problem ����� and problem������ respectively� Instead of problem ����� we consider the relaxed minimaxproblem

�����to �nd a pair �u� �� � V � �Q �K� such that

L�u� �� � L�u� �� � L�v� �� �v � V� � � Q �K������

where

L�v� �� � �Z

���F � u�d�

Z����u�� � � � �u� � v� � div� � g����� f � v�dx

is the Lagrangian of the relaxed problem�

V �nv � L

nn�� ���Rn� � sup

f��bg�D�A���k�k�kbkL���������Z

���b � u�d�

�Z

��� � ��u�� � �u� � v� � div� �dx � ��

o

Variational Methods �

is the extension of the set V� � u� of admissible displacement �elds� and

D�A�� �nf�� bg � �� L������R

n� � div� � Ln���Rn��

Z�

�� � ��v� � v � div�

�dx �

Z���

b � v d �v � V�o�

Q �n� � � � f�� Fg � D�A��

o�

As it was remarked above we have the inclusion

�����������V � BD���Rn� � fv � L����Rn� �

kvkBD��Rn� �Z�

jvjdx�Z�

j��v�j � ��g������

According to Theorem ����� problem ����� has at least one solution� Moreover�a pair �u� �� � V � �Q � K� is a saddle point of the minimax problem �����if and only if � � Qf � K is a maximizer of problem ���� and u � V is aminimizer of the following problem

!�u� � inff!�v� � v � Vg�������

where !�v� � supfL�v� �� � � � Q �Kg�

We also remark that !�v� � I�v� for any v � V� � u� and that

�Z�

�p�k� � g���t��dt � ��������

implies the validity of

I�v� � !�v� �v � V� � u��

We call all solutions to ����� weak solutions of the minimax problem ����� andall solutions to problem ������ weak solutions to the variational problem ������

� FUCHS et al�

Setting t� � supft � � g���t� �p�k�g we assume

�����g� is of class C

� on�� t�� t���

g��� � on�� t�� t���������

In this case the solution of problem ���� which has the clear physical mea�ning of the stress tensor is unique� In particular� the decomposition of ourelastoplastic body into elastic �F��� � �� and plastic �F��� � �� zones is alsounique�

In what follows we will assume that the conditions ������ ������ ���� providingthe existence of weak solutions to the minimax problem ����� and also theconditions ������ providing unique solvability of problem ���� with respect tothe stress tensor hold�

The main purpose of this chapter is the investigation of the di�erentiabilityproperties of weak solutions to problem ����� in the interior of �� For thisreason we need a local variant of the inequality

L�u� �� � L�u� �� for all � � Q �K�������

Let � � C�� ��� be an arbitrary cut�o� function satisfying � � � � � in ��

Next� let � be an arbitrary tensor�valued map from the set �n �K where

�s � f� � � � div � � Ls���Rn�g� s � ��������

We set � � � � ��� � �� and show that � � Q � K� It is clear that since� � � � � we have � � K� Taking into account that � and � � �n �div � ��f � Ln���Rn�� we get from Lemma A���

�� � � Ln��� M n�nsym ��� � � Ln��� M n�n

sym ��

and therefore

div � � div � � � div�� � �� � �� � ��r� � Ln���Rn��

Variational Methods �

Next� we consider the expression

�����������A �

Z�

�� � ��v� � v � div �

�dx �

Z�

h� � ��v� � v � div �

����v� � �� � �� � �v � div�� � �� � v � ��� � ��r��idx� v � V��

�����

Since � � Q we may write �keeping v �xed�

�����������A �

Z���

F � v d�Z�

h���v� � �� � �� � �v � div�� � ���

�v � ��� � ��r��idx� v � V��

�����

According to Lemma A���� �applied to � � � � �n� there exists a sequence�m � C���� M n�n

sym � such that

���������������

�m � � � in Ln��� M n�nsym ��

div �m div�� � �� in Ln���Rn��

�mD �� �� � ��D in L���� M n�n

sym ��

and thus �recall that v � D������ � Ln

n�� ���Rn��

Am �Z���

F � v d�Z�

h���v� � �m � �v � div �m � v � ��mr��

idx A

for any v � V��

On the other hand we may use integration by parts to see

Am �Z�

h���v��m � v � div���m�

idx�

Z���

F � v d �Z���

F � v d �v � V��

So A �R

���F � v d and ����� means that � � Q� Inserting � into inequality

������ and using convexity of g�� we get the appropriate local variant of ������

�Z�

h�u � div�� � �� � �u�r�� � �� � �� ���g����� g�����

idx � ����� �

� FUCHS et al�

which is valid for any � � �n �K and � � C����� such that � � � � � in ��

Here the pair �u� �� is an arbitrary weak solution to problem ������

Our approach to the local regularity of weak solutions is now based on theobservations that the stress tensor is uniquely determined and that the mostimportant physical characteristics are expressed in terms of this tensor� for ex�ample the distribution of elastic and plastic zones� For this reason it seemed tous that we should expect better regularity for stresses than for strains whichturned out to be true� So we �rst will try to get regularity for the stressesand then� using the variational inequality ���� �� we discuss regularity of thestrains� This is the crucial point of our approach� It is worth to remark thatall results on regularity for weak solutions to variational problems in perfectelastoplasticity which are known to the authors have been established in thisway �see �EK�� �Se�� � �Se���� �BF��� We do not touch the regularity problemfor twisting elastoplastic bars being treated for example in �BS�� �CR�� �F� withthe help of di�erent methods�

Let us brie�y describe the main results of this chapter� To avoid nonessentialtechnical di�culties we assume in addition that

u� � W �� ���R

n��������

Next� we suppose that the sets ��� and ��� are chosen in such a way that thefollowing density condition holds� namely

�����V� �W �

� ���Rn� is dense in V�

with respect to the norm of the space D�������������

As it is shown in the Appendix �see A��� the density condition ������ is satis��ed in particular if ��� � �� or ��� � ���

In what follows we will always assume that the conditions ������ ������ ������������ ������ and ������ hold and we will not mention this explicitly in thenext statements�

Our �rst result on the regularity of solutions to problem ���� is

Variational Methods

THEOREM ����� Suppose in addition to the above hypothesis that

jf j� j��f�j � L�loc���������

and let the function g� satisfy the condition

�c� � � g����t� � c�g���t��t for all � � t � t��������

Then we have weak dierentiability of the unique solution � of ����� i�e�

� � W ���loc��� M

n�nsym ��������

REMARK ����� Letting t tend to zero in inequality ������ and taking intoaccount conditions ������� we get c� � ��

REMARK ����� It is very natural for plasticity theory to assume that thefunction t g���t� is concave for t �� In this case condition ������ is satis�edwith c� � ��

In the case of strict convexity of the function g�� i�e� for t� � ��� we have

LEMMA ����� Suppose that the following two conditions hold�

the function t g���t� is concave for t � t� �������

and

t� � ���������

Then

jfx � � � F���x�� � �gj � �������

Now we can formulate the main result of this chapter�

FUCHS et al�

THEOREM ����� Suppose that the following conditions are satised�

f � W �n�loc���R

n� for some n n�����

and

���������������

for any � � ���p�k�� there is a constant H � such that

�g���j� j�j� j � � g���j�j�

j�j �

�� �� � �� � Hmax fg����j� j�� g

���j� j�j� j gj� � �j�

for all � � M n�nsym and all � � M n�n

sym such that g���j�j� � ��

���� �

Let � denote the unique solution of ����� Then there exists an open subset�� � � having the properties

� � C ���� Mn�nsym � for any � � ��� ��������

and� moreover�

F���x�� � � for all x � ���������

F��� � � a�e� in � n ���������

We note two important corollaries of Theorem ������ For any set �� �� �� weput d � supfj�D�x�j � x � ��g� By Theorem ������ d �

p�k�� Let us use the

variational inequality ���� � choosing a cut�o� function � so that � � � outsideof �� and � � � on ��� Then we put � � � � �� where � � C�

���� Mn�nsym ��

j� j � � outside of ��� j� j � � in �� and the number � satis�es the condition� � � � �

p�k� � d���� It is clear that � � �n � K and so inequality ���� �

implies the estimate

�Z��

u � div� dx � �

Z��

hg��� � ���� g����

idx�������

Variational Methods

On the sets f� � Mn�nsym � j�Dj � t�g and � � t�� t�� the conjugate functions of

g and g� coincide with their Legendre transformations and therefore we have

�����������������������������������������

g���� � � � � � g���� � � �g��

���� j�Dj � t��

or

g��� � � � � � g����� � � �g���

���� F��� � ��

and

g���s� � st� g��t�� s � g���t�� jtj � t��

or

g��t� � st� g���s�� t � �g�����s�� jsj � p

�k��

������

Thus letting � � in ������� we obtain

��u� ��g�

����� or � �

�g

�����u�� in ���

We summarize the above discussion in the following assertion�

COROLLARY ����� Let u � V be a weak solution to the variational pro�blem ������ i�e� the pair �u� �� � V��Q�K� is a saddle point of the Lagrangianon the set V � �Q � K�� Then the strain tensor ��u� is H�older continuouson ��� and moreover the displacement vector u satises in �� the followingsystem of dierential equations of elliptic type�

div� �g��

���u���� f � ��

Thus the higher�order regularity of u on �� can be established using a well�known scheme �see �Gi��� and it is determined only by the properties of g� andf �

COROLLARY ����� Assume that the hypotheses of Theorem ����� and Lem�ma ����� hold� Then j� n ��j � �� and we have partial regularity of the weaksolution� that is regularity on an open set of complete measure�

At the end of this section we demonstrate a few simple criteria for checkingcondition ���� ��

� FUCHS et al�

LEMMA ����� Condition ���� � is a consequence of ������ and the inequality

��� t

g���t�� s

g���s�

��� � jt� sjp�k�

� t� s ��������

Proof� We have for all � � ���p�k�� and any ��� � M

n�nsym such that

g���j�j� � �

�g���j� j�j� j � � g���j�j�

j�j �

�� �� � �� �

g���j� j�j� j j� � �j��

��g���j� j�j� j � g���j�j�

j�j�� � �� � �� � g���j� j�

j� j j� � �j�

�g���j� j�j� j

g���j�j�j�j j�jj� � �j

��� j�jg���j�j�

� j� jg���j� j�

���� g���j� j�

j� jnj� � �j� � j���j�p

�k�

o���� p

�k�

��

g���j� j�j� j j� � �j� �

��� p

�k�

��c� max

ng����j� j�� g���j� j�

j� joj� � �j��

Thus we can take

H ��

c�

��� �p

�k�

and Lemma ����� is proved��

It is easy to check that conditions ������ and ������ hold for the integrand

g��t� �

��������t�� jtj � t� �

k�p��

k��p

�jtj � k���

�� jtj t��

describing Hencky�Il�yushin plasticity and therefore all hypothesis of Theorem����� are satis�ed� In this model it is quite natural to call the domain �� theelastic zone since in this domain the plastic part of the deformation is equalto zero and� by Corollary ������ the displacement �eld u is governed by thesystem of PDE�s from the theory of linear elasticity�

Variational Methods �

We note also that for � � t � t� condition ������ is equivalent to the followingone

���� t

g���t�

����� � �p�k�

� � � t � t��������

In particular� for the integrand

g��t� �p�k��

p� � t� � ��

we have t� � ��� the function

t g���t� �

p�k�tp� � t�

is concave for t � and moreover

g����t� �p�k�

�� � t����� g���t�

t� � �

� t

g���t�

���

�p�k�

tp� � t�

� �p�k�

for t �� and so conditions ������� ������ for c� � �� ������ for any t� � hold�Thus we can apply Theorem ����� and Corollary ����� which provide partialregularity of any weak solution�

� FUCHS et al�

��� Approximation and proof of Lemma �����

We will study the case ��� � �� The case ��� � � requires some non�essentialmodi�cations and we recommend the reader to carry out the details� Weconsider a family of variational problems depending on a parameter � ���� ��

�����to �nd u� � V� � u� such that

I��u�� � inffI��v� � v � V� � u�g�

�����

where

V� � V� �W �� ���R

n�� I��v� ��

Z�

j�D�v�j�dx� I�v��

Problem ����� has a unique minimizer u� � V� � u� which satis�es a nonlinearsystem of PDE�s of elliptic type

��������� � ��D�u�� � �g

�����u��� �

� ��D�u�� �K� div u��� g���j�D�u��j� �D�u��j�D�u��j �

�����

Z�

�� � ��v�dx � M�v� �Z�

f � v dx�Z���

F � v d �v � V�������

and therefore

div �� � f � � a�e� in �������

We have

LEMMA ����� For any � ���� �� the following estimate is true

p�k�D�u��kL���� � kdiv u�kL�����

�k�D�u��kL���� � ku�kL nn�� ���

� C����

where the positive constant C depends only on

kfkLn���� kFkL������� ku�kW �� ��R

n�

Variational Methods �

and �� � being dened in �����

Moreover� there exists a subsequence of the sequence �u�� ���� which will be de�noted in the same way� such that

����� u� � u in Ln

n�� ���Rn��

����� u� u in Lr���Rn� for r � ��� nn�� ��

�����Z�

� � ��u��dxZ�

� � ��u� �� � C�� ��� M n�n

sym ��

����� div �� � div � in L�����

����� �Z�

j�D�u��j�dx ��

��� � �� � � in L���� M n�nsym ��

����� ��D � ��D�u���� �D in L���� M n�n

sym ��

where � is the unique solution to problem ����� and u is a solution to problem�������

Proof� Let s � denote the solution of the equation

g���s� �p�k�

� � �

����� �

Using estimate ������ of the previous chapter� we get

�� I��u�� � I��u�� � I��u�� � �

Z�

j�D�u��j�dx

���K�

Z�

div�u�dx�p�k�

�� �

Z�

j�D�u��jdx�

� �n

Z�

jtr��j jdiv u�jdx�Z�

�� � ��u��dx�M�u���

��g��s��

p�k� �� s

�j�j�

� FUCHS et al�

From this and also from the embedding theorem A���� we deduce estimate�����

It follows from ����� that the sequences f��g and f��D � ��D�u��g are boun�ded in L���� M n�n

sym � and L���� M n�nsym �� respectively� Therefore� we can state�

in view of ����� that the claims ����� � ������ ��� � and ����� have beenestablished� It remains to prove that u and � are solutions to problems ������and ����� respectively� and that ����� holds�

Since � � � �����D�u�� � K and since the setK is weakly closed in L���� M n�nsym ��

it follows that � � K� Now passing to the limit in ����� and using the densitycondition ������� we �nd that � � Qf �

On the other hand� the duality relations imply that

� � � ��u��� g���u���� g��� �� � � a�e� in ��

But then� by ����� and ������ we have

I��u�� � �

Z�

j�D�u��j�dx �Z�

�� � � ��u��� g��� ��

�dx�M�u��

� � ��

Z�

j�D�u��j�dx�Z�

�� � ��u��dx�Z�

g��� ��dx�M�u��

� � ��

Z�

j�D�u��j�dx�Z�

g��� ��dx�Z�

�� � ��u��dx�M�u���

Taking into account the statements of Theorem ����� of the previous chapter�we get

supfR��� � � � Qf �Kg � inffI�v� � v � V� � u�g � I�u��

� I��u�� � � �

Z�

j�D�u��j�dx�Z�

��� � ��u��� g��� ���dx�M�u���

Passing to the limit in this inequality� using the upper semicontinuity of thefunctional

� �Z�

g����dx

Variational Methods �

with respect to the weak�� topology of � and the fact that � � Qf � K� weobtain

supfR��� � � � Qf �Kg � R��� � lim sup���

���

Z�

j�D�u��j�dx������

Inequality ����� implies that � is a maximizer of problem ���� and also that

lim sup���

h� �

Z�

j�D�u��j�dxi� ��

By passing to a subsequence �if necessary� we deduce claim ����� togetherwith the identity

lim���

I�u�� � inffI�v� � v � V� � u�g

which shows that u� is a minimizing sequence of problem ������ and thereforeit converges weakly in L

nn�� ���Rn� to a solution of problem ������� Lemma

����� is proved��

Proof of Lemma ������ Let �� � maxft�� j�D�u��jg� Estimate ���� impliesthat there exists a bounded positive Radon measure � such that

Z�

���dxZ�

�d� �� � C�� ����

Let Q be a closed cube in �� Then

lim sup���

ZQ

��dx � ��Q�������

Further we have �recall � � � �� � � �D�u�� and equation ������

ZQ

j� �Djdx �ZQ

g���j�D�u��j�dx �ZQ

g������dx�

� FUCHS et al�

From condition ������ and Jensen�s inequality we infer the estimate

Z�Q

j� �Djdx � g��� Z�Q

��dx��������

Taking into account ������ ������ and the results of Lemma ����� we obtain

�����������������������

Z�Q

j�Djdx � lim inf���

Z�Q

j� �Djdx �

� lim inf���

g���Z�Q

��dx�� g��

�lim sup���

Z�Q

��dx��

� g�����Q�jQj

��

������

Let x � � be the center of the cubes Q� The function

x d�

dx�x� � lim

jQj��

��Q�

jQj

is summable with respect to Lebesgue measure in Rn � see �DS�� and is therefore�nite almost everywhere with respect to this measure� But ������ implies that

j�D�x�j � g���d�dx

�x��for a�a� x � ��

and by ������ j�Dj � p�k� a�e� in �� Lemma ����� is proved�

Variational Methods

��� Proof of Theorem ����� and a local esti�

mate of Caccioppoli type for the stress

tensor

We denote by V� the �nite�dimensional space of rigid displacements� i�e�

V� � fv� � a� �x � a � Rn � � is a skew symmetric tensorg�

We continue to consider the regularized problem ������ Since f � Ln���Rn��it is easy to show� using the method of �nite di�erences� that

��u��� �� � W ���loc��� M

n�nsym �������

and therefore we have

Z�

���k � ��v�dx � �Z�

f � v�kdx �v � C�� ���Rn�� k � �� �� � � � � n������

so that

��ij�j � �fi a�e� in �������

Let us introduce the bilinear forms E�� � �� �� and E��� � �� �� on the space M n�nsym

by de�ning

���������������������

E�� � ���� ����g����

�j� j���� � �

�g���j� j�j� j � � � �

�g����j� j�� g���j� j�

j� j�� �� � ��j� j� �

E��� � ���� � ��D � �D �K�tr� tr� � E�� � �D��D��

�����

where � � M n�nsym is a tensor parameter� Condition ������ provides the estimates

h��j� j�j�j� � E�� � �� �� � h�j� j�j�j� � c�g���j� j�j� j j�j������

FUCHS et al�

in which

h��t� � minfg����t��g���t�tg� h�t� � maxfg����t��

g���t�tg� t ��

Moreover� since

���k � ��D�u��k� �K�div u��k �

��g��� �

�j�D�u��j��D�u��k������

it follows that

���k � � � E���D�u��� �D�u��k���� �� � M n�nsym ���� �

Now we are going to derive some inequalities for r��� We have �using sum�mation w�r�t� k�

�������������������������������������

jr��j� � ���k � ���k � E���D�u��� ��u��k�� ��k�

��E���D�u��� ��u��k�� ��u

��k��

����E���D�u��� ���k� �

��k����

� ����k � ��u��k��

��h�jr��Dj� �K�jrtr��j�

�c�g���j�D�u��j�j�D�u��j jr��Dj�

i���

�����

We put

�� � supfg���t��t � t �g�����

and show that �� � ��� Indeed� let us take and �x some t� ���� t��� Then�by condition ������� we have

�� � max f supfh�t� � t � ��� t��g� supfg���t�

t� t t�gg�

where h � C����� t��� is de�ned according to

h�t� �

�����g���t��t if t �

g������ if t � ��

Variational Methods

Thus

�� � max f supfh�t� � t � ��� t��g�p�k�t�

g � ���

Since � ���� ��� it follows from ����� that

jr��j� �����k � �

D�u��k����h

�� � ��c��jr��Dj� �K�jrtr��j�i��

� c��c�� ��� K������k � ��u

��k��

��jr��j�

So we have established the estimate

jr��j� � c�����k � ��u

��k��������

Note that in order to derive estimate ������ we made essential use of condition������ of Theorem ������

Let us introduce some additional tensor�valued and vector�valued �elds asfollows

�� � �� � ��� u� � u� � ���x� x��� v��������

where �� and �� are arbitrary matrices from Mn�nsym � and v� is an arbitrary rigid

displacement from V�� Next� let � � C����� be an arbitrary cut�o� function�

By ����� we can apply ����� to the function

v � � u��k

and get

���������������

Z�

���k � ��� u��k�dx �

Z�

f�k�� u��k�dx �

� �Z�

f � � "u�dx�Z�

f � � �ku

��kdx�

������

Since

�"u� � div ��u��� �

�rdiv u��

� FUCHS et al�

it follows that

�����������������������������������������������������������������������������������

�Z�

f � � "u�dx�Z�

f � � �ku

��kdx

� ��Z�

f � � div ��u��dx �Z�

f � � rdiv u�dx�Z�

f � � �ku

��kdx

� �Z�

��f� � � ��u��dx� �Z�

�f �r� � � ��u��dx

�Z�

� f � rdiv u�dx�Z�

f � � �ku

��kdx

� �Z�

� ��f� � ��u��dx�Z�

� f � rdivu�dx�Z�

fi� �ju

�j�idx

� �Z�

� ��f� � ��u��dx�Z�

� f � rdiv u�dx

�Z�

r� � u�div f dx�Z�

�f � u�� � r�� dx�

������

So ������ and ������ give us the decomposition

J �Z�

� ���k � ��u��k�dx � J� � J� � J��������

where

�������������������������������������

J� � ��Z�

��ij�k� �i�jk�u

��dx�

J� �Z�

��ij�k� �iu

�k�jdx�

J� �Z�

h�� ��f� � ��u�� � � f � rdiv u�

�r� � u�div f � �f � u�� � r�� idx�

�����

Variational Methods �

Next we use the formulas

��u�� � �D�u�� ��

ndivu��� �� � ��D �

ntr ���

and identity ������ As a result we have the relation

�������������������������������������������������������������������������������������

J� � ��Z�

��ij�k� �i�

Djk�u

��dx� �

n

Z�

��ij�j� �idivu

�dx

� � �n

Z�

tr���k� �i�

Dik�u

��dx� �Z�

��Dij�k� �i�

Djk�u

��dx

� �n

Z�

f � r� divu�dx � �Z�

�fk � ��Dks�s

�� �i�

Dik�u

��dx

��Z�

��Dij�k� �i�

Djk�u

��dx��

n

Z�

f � r� divu�dx

� �Z�

�f �r� � � �D�u��dx��

n

Z�

f � r� divu�dx

���Z�

����Dij�k�� ��i�

Djk�u

�� � �ik��s�Djs�u

���dx

� �Z�

�f �r� � � ��u��dx� ��Z�

����D�k � S�k�dx

�����

where the matrices S�k�� k � �� �� � � � � n� are given by

S�k� ���ik��s�

Djs�u

��� ��i�Djk�u

���� �ij �

������� i � j

�� i � j�

We note the following relations

S�k� � S�k� � c��n�jr�j�j�D�u��j��

��D�k � S�k� � ����D�k �

�S�k� � �S�k��T

�� �

�jS�k� � �Sk�T j� � E

��D�u��� �D�u��k�� S

�k��

� FUCHS et al�

which together with ������ ����� ���� imply

��D�k � S�k� � jr�jpc�h� � c�

g���j�D�u��j�j�D�u��j

i��j�D�u��jh��D�k � �D�u��k�i��

Thus ����� and the last inequality give the bound

���������������

J� � �Z�

�f �r� � � ��u��dx�

���pc�J

���Z�

��jr�j�h� � c�

g���j�D�u��j�j�D�u��j

ij�D�u��j�dx

����

���� �

Let us transform J�� using integration by parts� in the following way

J� � �Z�

���ij� �iku

�k�j � ��ij�

�jdiv u

��i�dx �

�Z�

��ij�j� �iku

�kdx �

Z�

��ij� �ijku

�kdx

�Z�

��ij�j� �idivu

�dx�Z�

��ij� �ijdivu

�dx

� �Z�

�f � u�� � r�� dx�Z�

f � r� div u�dx

�Z�

��ij� �ijku

�kdx�

Z�

�� � r�� divu� dx�

Combining this with ������ ���� � gives�

���������������

J � ��pc�J

���Z�

��jr�j�h� � c�

g���j�D�u��j�j�D�u��j

ij�D�u��j�dx

���

�Z�

��ij� �ijku

�kdx�

Z�

�� � r� divu�dx� J��

������

Variational Methods �

where

���������������������������

J� � �Z�

h�f �r� � � ��u�� � � ��f� � ��u��

�f � r� div u� � �f � u�� � r�� idx�

�Z�

�� div f div u� �r� � u�div f�dx�

������

After application of Young�s inequality we �nally get

���������������

J � ���c�

Z�

��jr�j�h� � c�

g���j�D�u��j�j�D�u��j

ij�D�u��j�dx

��Z�

��ij� �ijku

�kdx� �

Z�

�� � r�� divu�dx� �J��

������

After these preparations we are ready to give the

Proof of Theorem ������ We put �� � �� � �� v� � � in ������� In view of

����� we have

div u� ��

nK�

tr �� in ��������

hence it follows from ������� ������ and ����� that

���������������

�c��

Z�

� jr��j�dx � ���c�

Z�

��jr�j�h�j�D�u��j� � c�

p�k�j�D�u��j

idx

��J� ��

nK�

Z�

tr �� �� � r�� dx� �Z�

��ij� �ijku

�kdx�

������

Now let us estimate the integral

J� � �Z�

��ij� �ijku

�kdx � �

Z�

����ijAijku�kdx

� FUCHS et al�

where

Aijk � n����i��j��k � ����i��jk � ��j��ik � ��k��ij�� � ����ijk

o�

During the following calculation we will use the restriction n � � or �� By thisand the embedding theorem we have the inequality

�Z�

j����jndx��n � c��n�d

�Z�

jr������j�dx���

� p�c�d

h�Z�

� jr��j�dx���

� ��Z�

��jr�j�j��j�dx���i

d � jspt�j �n� �� �

and thus

J� � c��n��Z�

j����jndx��n�Z

�Aijku�kAijmu

�m�

n��n���dx

�n��n

� c��n�dh�Z

� jr��j�dx���

� ��Z�

��jr�j�j��j�dx���i�

�Z�

�Aijku�kAijmu

�m�

n��n���dx

�n��n �

So ������ can be rewritten in the form

�c��

Z�

� jr��j�dx � ���c�

Z�

��jr�j���j�D�u��j� � c�

p�k�j�D�u��j

�dx

��J� ��

nK�

Z�

tr���� � r�� dx� c�dh�Z

� jr��j�dx���

���Z�

��jr�j�j��j�dx���i�Z

�Aijku�kAijmu

�m�

n��n���dx

�n��n �

Variational Methods �

It remains to apply Young�s inequality in order to get the �nal estimateZ�

� jr��j�dx � c �n�K�� k�� c�� c��

hZ�

h��jr�j���j�D�u��j� � j��j� � j�D�u��j�

�jr�� jj��j�idx � jJ�j� d��Z�

�Aijku�kAijmu

�m�

n��n���dx

� ��n���n

i

� c��n�K�� k�� c�� c�� ��hZ�

��j�D�u��j� � j�D�u��j� j��j��dx�

��Z�

ju�j nn��dx

� ��n���n �

�kfkL��spt�� � k��f�kL��spt��

�Z�

j��u��jdx�

��kfkL���� � kdiv fkL�spt��

�ktr��kL�����

��kfkLn��� � kdiv fkLn�spt��

�ku�k

Ln

n�� ���

i�

Taking into account the hypotheses of Theorem ����� �see �������� estimate���� of Lemma ����� and relations ������ we derive from the last estimate that

kr��kL����� � c����� ��� b ��������

Letting � � and using Lemma ������ we have established ������� Theorem����� is proved�

LEMMA ����� �Local estimate of Caccioppoli�type�� Let all conditions ofTheorem ����� hold� Then there is at least one pair �u� �� � V � �Q � K�which is a weak solution to the minimax problem ����� and which satises theestimate�����������������

�Rn��

ZBsR�x��

jr�j�dx �c��n�K�� k����� ��� kfkW �

n���Rn��

��� s��H�

nR� �

h� Z�

BR�x��

j�jrdx��r

��

R

� Z�

BR�x��

jujr�dx��r�i�o������

� FUCHS et al�

being valid for all BR�x�� b �� b �� � � R � � and s ���� ��� Here r ��n� �nn�� ��

r � n� r� � rr�� � � � � � ��� u � u� �

��x � x��� v�� where �� and �� arearbitrary tensors from M

n�nsym satisfying the relations

�������tr�� � nK�tr�

��D �g���j��Dj�j��Dj �

�D

������ �� ��g

������� �

� ��g�

������������

j��Dj � �� �p�k������

and v� denotes some rigid displacement�

Proof� We �rst note that ���� � implies ������� Indeed� let t � and let usput � � �� � � k�� j� j � t� Then we have

tg���t� � Hk� maxfg����t��g���t�tgt�

and therefore

g���t�t

� Hk�g����t� �t � �� c� �

Hk������� �

Thus we can state that all conditions of Theorem ����� hold� and we haveestimate ������ since condition ����� implies f � W �

��loc���Rn�� Let us go

back to estimate ������� The �rst identity in ����� allows us to write

divu� ��

nK�tr��������

where

�� � �� � ��� u� � u� � ���x� x��� v��

and therefore

�Z�

�� � r�� divu�dx � c���n�K��Z�

jr�� j j��j�dx�������

Variational Methods

We integrate by parts in the �rst two terms of ������ and use identity ������to get

���������������������������������������������������

J� � ��Z�

u� � div�� ��f� � f �r� �dx

� �nK�

Z�

tr ����f � r� � � div f �dx

�Z�

���f � u�� � r�� �r� � u�div f �dx

� c���n�K��Z�

nju�j�� jdiv ��f�j� jr� j jrf j� jf j jr�� j�

�j��j�� jrf j� jr� j jf j�odx�

������

It follows from ������� ������ and ������ that

���������������������������������������

J � ���c�

Z�

��jr�j��� �

Hk�

g���j�D�u��j�j�D�u��j

�j�D�u��j�dx

�c���n�K��Z�

jr�� j j��j�dx� c���n�Z�

j��j jr�� j ju�jdx

��c���n�K��Z�

nju�j�� jdiv ��f�j� jr� j jrf j� jf j jr�� j�

�j��j�� jrf j� jr� j jf j�odx�

������

Let

J� � �Hk�

Z�

g���j�D�u��j�j�D�u��j ��jr�j�j�D�u��j�dx

� �Hk�

Z�

h�j�D�u��j���jr�j�j�D�u��j�dx

where

h�t� � max fg����t��g���t�tg� t � ��

FUCHS et al�

We now make use of the variational identity ����� by setting v � #u� with acut�o� function # � C�

� ���� As a result we get the equationZ�

#�� � ��u��dx �Z�

#f � u�dx�Z�

�� � �u� �r#�dx�������

Taking into account the identities ������ ������ we can write

�� � ��u�� � K�div�u� � �j�D�u��j� � ���D � �D�u���

�g���j�D�u��j�j�D�u��j �D�u��� g���j��D j�

j��Dj ��D�� ��D�u��� �

�D�

and after application of condition ���� � we arrive at the estimate

�� � ��u�� � ���D � �D�u�� �H�h�j�D�u��j�j�D�u��j��Inserting # � ��jr�j� into ������ and using the last estimate� we get

J� � �Hk�H��

nZ�

��jr�j�f � u�dx�Z�

�� � �r���jr�j��� u��dx

��Z�

��jr�j���D � �D�u��dxo�

From the latter inequality and also from ������ and ������ it follows that�������������������������������������������������������������������������������

�c��

Z�

� jr��j�dx � ���c�

Z�

��jr�j��j�D�u��j�dx�

�c���n�K��Z�

jr�� jj��j�dx�

�Z�

j��jju�j� ���c�Hk�H�

jr���jr�j��j� c��jr�� j�dx

� ���c�Hk�H��

�Z�

��jr�j�j��Djj�D�u��jdx

� ���c�Hk�H��

Z�

��jr�j�jf jju�jdx

��c���n�K��Z�

nju�j�� jdiv ��f�j� jr� jjrf j�

�jf jjr�� j� � j��j�� jrf j� jr� jjf j�odx�

������

Variational Methods

As remarked earlier condition ����� implies f � W ���loc���R

n�� Taking intoaccount estimate ���� �� we see that all hypotheses of Theorem ����� hold� andthus we have estimate ������� By the embedding theorem and Lemma �����we further have�����

�� � in Lr���� Mn�nsym �

u� u in Lr����Rn�������

where the pair �u� �� � V��Q�K� is a weak solution to the minimax problem������ Passing to the limit in inequality ������ and using the statements ofLemma ����� we get

�������������������������������������������

Z�

� jr�j�dx � c���u�K�� k�� c��H�

nZ�

jr�� jj�j�dx�

�Z�

j�jjuj�jr���jr�j��j� jr�� j�dx�

�Z�

juj�� jdiv ��f�j� jr� jjrf j� jf j���jr�j� � jr�� j��dx�

�Z�

j�j�� jrf j� jr� jjf j�dxo�

�����

Let the cut�o� function � satisfy the conditions

� � � � � in �� spt � � BR�x�� b �� b �� � � � in BsR�x��� � � s � ��

jrk�j � constRk���s�k � k � �� �� ��

Then estimate ����� implies that

Rn��

ZBsR�x��

jr�j�dx � R� c���n�K�� k�� ���H�

n �

R���� s��

Z�

BR�x��

j�j�dx

� FUCHS et al�

��

R���� s��

Z�

BR�x��

j�jjujdx

�Z�

BR�x��

jujhjdiv ��f�j� �

R��� s�jrf j� jf j �

R���� s��

i

�Z�

BR�x��

j�j�jrf j� �

R��� s�jf j�dx

o� c���n�K�� k�� ���

H���� s��

n Z�

BR�x��

j�j�dx

��

R

� Z�

BR�x��

j�jrdx��r� Z

�BR�x��

jujr�dx��r�

�RkfkW �����Rn�

h Z�

BR�x��

j�jdx��

R

Z�

BR�x��

jujdxi

�R�� Z�

BR�x��

jujr�dx��r�� Z

�BR�x��

jdiv ��f�jrdx��ro �

� c��H���� s��

n� Z�

BR�x��

j�jrdx��r

��

� Z�

BR�x��

j�jrdx��r

��

�R�

� Z�

BR�x��

jujr�dx��r�

�RkfkW �����Rn�

h� Z�

BR�x��

j�jrdx��r

��

R

� Z�

BR�x��

jujr�dx��r�i

��

R

� Z�

BR�x��

jujr�dx��r�� Z

�BR�x��

jdiv ��f�jndx� �nR�

o�

Variational Methods �

� c����n�K�� k�� ���H���� s��

nh� Z�

BR�x��

j�jrdx��r

��

R

� ZBR�x��

jujr�dx��r�i�

�R�kfk�W �����Rn��R���n

n���Z��

jdiv ��f�jndx� �no�

�c� �n�K�� k�� ������ kfkW �

n���Rn��

H���� s���

nR� �

h� Z�

BR�x��

j�jrdx��r

��

R

� Z�

BR�x��

jujr�dx��r�i�o

It remains to put c� � c� � and we are done� Lemma ����� is proved��

�� FUCHS et al�

��� Estimates for solutions of certain systems

of PDE�s with constant coecients

Suppose that we are given functions

� � Lr�B� M n�nsym � and v � Lr��B�Rn��

where r � and r� � rr�� � satisfying the following system of partial di�erential

equations

�������v� � ��g�

�������� �

div � � ������

on the unit ball B � Rn � Here the tensor �� � M n�n

sym is chosen according to

j��Dj � � �p�k�������

LEMMA ����� For any � � ���p�k�� and any t� ���� �� a constant c�� �

c���t�� �� exists such that

W �t� � c��tW ��������

for all � � t � t��

In Lemma ����� we used the notation

W �t� ��Z�Bt

j� � ���tjrdy��r

��t

infv��V�

�Z�Bt

jv � ��g�

�� ��������ty � v�jr�dy

��r��

Bt denoting the ball of radius t concentric to B and ���t �R�Bt

� dy is the mean

value of � w�r�t� Bt�

Variational Methods ��

Proof� We put �� � �g���

����� By condition ����� we have

��g

������� �

���g��� �

�������

� M n�nsym M

n�nsym ������

and therefore system ����� is equivalent to the system of linear elasticity

�����div � � � �

� � ��g���

������v�

in B�����

Let

E���� �� ��� �

� ��g���

������ �� � K�tr � tr �

� � E��D� �D� ��D�

��� �� �� � M n�nsym �

Here the bilinear form E is de�ned by formula ������ i�e�

E�� � ���� ����g����

�j� j���� � �

�g���j� j�j� j � � � �

�g����j� j�� g���j� j�

j� j�� �� � ��j� j� �

Clearly the form E satis�es the estimates ����� i�e�

h��j�Dj�j�Dj� � E��D� �D� �D� � h�j�Dj�j�Dj��

From condition ����� we deduce that h��j��Dj�� h�j��Dj� are positive and�nite� Thus� by letting

�� � minfK�n� h��j��Dj�g� � � maxfK�n� h�j��Dj�g�

we �nd that

��j�j� � E����� �� �� � �j�j� �� � M n�n

sym �����

The ellipticity estimate ���� guarantees that the functions v and � are in�ni�tely di�erentiable inside of B� Let us rewrite ���� as

ZB

E���� ��v�� ��w��dx �ZB

� � ��w�dx � � �w � C�� �B�Rn����� �

�� FUCHS et al�

where � � � � ����� We also put v � v� ���v���y�v� for a rigid displacementv� which is chosen according to �see Lemma ������Z

B

jvjr�dy � inf�v��V�

ZB

jv � ��v��y � v�jr�dy�

Remarking that

���v��t ���g�

����������t� � � t � �������

we get

� ���g

���������v������

and

v � v � ��g�

�� ����������y�� v��������

Let us take a cut�o� function � � C���B� having the properties

������ � � � � in B� � � � in Bt� � � � t� � t� � ��

jr�j � ���t� �

������

If we insert w � �v into ��� �� we arrive at the identity

Z�

�� � ��v�dy � �Z�

� � �r�� v�dy�������

From ������ ������ and ���� it follows that

ZBt�

j��v�j�dy � �

����� t��

ZB

jvjj� jdy�������

Further we haveZBt�

��k � ��w�dy � � �w � C�� �Bt� �R

n��������

Variational Methods ��

Now we take � � C�� �Bt�� so that � � � � � in Bt� � jr�j � �

t��t� in Bt� � � � �

in Bt� � where t� ��t�� t��� Inserting w � �� v�k into ������ with v � v � w��w� � V�� we get

ZBt�

����k � ��v�k�dy �Z

Bt�

��E����� ��v�k�� ��v�k��dy �

� ��Z

Bt�

���k � � v�k �r��dy � ��Z

Bt�

�E����� ��v�k�� v�k �r��dy

� ��Z

Bt�

��E����� ��v�k�� ��v�k��dy

����Z

Bt�

E����� v�k �r�� v�k �r��dy

���

and by ����

��Z

Bt�

jr��v�j�dy �Z

Bt�

��E����� ��v�k�� ��v�k��dy �

� �Z

Bt�

E����� v�k �r�� v�k �r��dy � ��

�t� � t���

ZBt�

jr vj�dy�

Choosing the rigid displacement w� in a suitable way� we �nd

ZBt�

jr vj�dy � c���n�ZBt�

j��v�j�dy�

and therefore

ZBt�

jr��v�j�dy � �� c�����t� � t���

��

�� t�

ZB

jvjj� jdy�

After a �nite number of such steps we can apply Sobolev�s inequality and arriveat

supBt�

jr��v�j� � c���t�� ��Z�B

j� jjvjdy�

�� FUCHS et al�

Now� for t � t�� we can write

�Z�Bt

j� � ���tjrdy��r

��Z�Bt

j ��g

����������v�� ���v��t�jrdy

��r �� c�����

�Z�Bt

j��v�� ���v��tjrdy��r �

c���n� �� r�t�Z�Bt

jr��v�jrdy��r � c��t

pc���Z�B

j� jjvjdy���

Further� by Korn�s inequality� we have

�t

infv��V�

�Z�Bt

jv � ���v��ty � v�jr�dy��r�

� c���n� r��Z�Bt

jr��v�jr�dy��r� � c��t

pc���Z�B

j� jjvjdy���

Now� from ����� and the last two estimates it follows that

W �t� � c���t�� ��t�Z�B

j� jjvjdy��� � c��t

�Z�B

j� jrdy���r�

�Z�B

jvjr�dy���r� � c��

t

h�Z�B

j� jrdy��r

��Z�B

jvjr�dy��r�i

� ��c��tW ����

and Lemma ����� is proved��

Variational Methods �

�� The main lemma and its iteration

Let us introduce another bilinear form

E��� � ���� ����g�����

�j� j���� � �

��g�� �

��j� j�j� j � � � �

��g���

���j� j�� �g�����j� j�j� j

�� �� � ��j� j� �

���� � � M n�nsym � j� j � p

�k��

For the corresponding quadratic form we have the bounds

h���j� j�j�j� � E��� � �� �� � h��j� j�j�j�����

where

h���s� � minn�g�����s�

s� �g���

���s�o� h��s� � max

n�g�����s�s

� �g������s�

oand jsj � p

�k��

Since the function g�� �� � p�k��

p�k�� R is the conjugate function of g� �

�� t�� t�� R �see formulas ������� we have

�g������s� �

g����t�for s � g���t� �� t � �g���

��s���

and therefore

h��g���t�� � max

n t

g���t��

g����t�

o� jtj � t��

If condition ������ holds� then we have the estimate

h��s� � h��g���t�� � maxf�� c�g �

g����t��

c�g����t�

� jtj � t��

and thus

h��s� � c��g������s�� jsj �

p�k������

� FUCHS et al�

MAIN LEMMA ����� Suppose that all the conditions of Theorem �����hold� Let the number r satisfy

r ��n� �n

n� ��� r � n�����

and let r� � rr�� � Moreover� x �� �� � and some numbers �� ���� ���

t� ���� ���

Let �u� �� � V � �Q � K� denote the weak solution of the minimax problem����� which is produced via the approximation procedure from section �� Lemma������ Then� for any t ���� t�� and � � ���

p�k��� there are positive numbers

�� � ���t� t�� �� ���� R� � R��t� t�� �� ���

such that the conditions

BR�x�� b �������

j��D�x��Rj � �����

U�x�� R� �R� � ������

for some ball BR�x��� � � R � R�� imply the decay estimate

U�x�� tR� � �c���t�� ��t�U�x�� R� �R����� �

Here c�� is the constant of Lemma ����� and the excess U�x�� R� is dened bythe formulas

U�x�� R� � U�x�� R� � U�x�� R��

U�x�� R� �� Z

�BR�x��

j� � ���x��Rjrdx��r

���x��R �Z�

BR�x��

� dx�

U�x�� R� ��

Rinf

v��V�

� Z�

BR�x��

ju� ���x��R�x� x��� v�jr�dx��r�

���x��R ��g�

������x��R��

Variational Methods �

Proof� We note �rst that �u� �� � V � �Q � K� satis�es the variationalinequality

�����������������������

�Z

nu � div���� � ��� � ��g����� g�����

ody �

� �Z

n�u � div�� � �� � �� � �� �

�u�r�� � ��g����� g�����ody � �

����

for all � � �n �K and all � � C����� such that � � � � ��

Suppose that the claim of the lemma is false� Then there exist numbers t� �and sequences xh� Rh� �h such that�

BRh�xh� b �������

�h � U�xh� Rh� �R�h � as h ��������

j��D�xh�Rhj � �������

U�xh� tRh� �c��t�U�xh� Rh� �R�h � � �c��t�h������

From ���� we get

���������R

BRh

�xh�n� uh � div���� � ��� � �����x

h�Rh � �� � ���

�g���� � g�����odx � �

�����

being valid for all � � C��BRh�xh��� M n�n

sym � and � � C���BRh�x

h�� such that

j�Dj � p�k� and � � � � � where we have abbreviated

���������������������

uh � u� ���xh�Rh�x� xh�� uh�� uh� � V�

���xh�Rh � �g�

������xh�Rh��

�h � � � ���xh�Rh�

�����

�� FUCHS et al�

Here the rigid displacement uh� is chosen according to

ZBRh �x

h�

juhjr�dx � infv��V�

ZBRh�x

h�

ju� ���xh�Rh�x� xh�� v�jr�dx�����

Now we use Lemma ����� with x� � xh� R � Rh� �� � ���x

h�Rh� As a resultwe have the estimate

Rn��h

ZBsRh �x

h�

jr�hj�dx � c������ f� ��

��� s��

nU��xh� Rh� �R�

h

o�����

Let us introduce the coordinate transformation x � xh � Rhy� y � B� andde�ne

vh�y� ��

�hRhuh�x�� �h�y� � �h�x���h���� �

It is easy to check that

U�xh� � � �hWh� �Rh�������

where

Wh�t� �

� Z�Bt

j�h � ��h�tjrdy��r

We have

U�xh� � � ��

infv��V�

� Z�

B��xh�

juh � ���xh�Rh�x� xh� � uh��

����xh���x� xh�� v�jr�dx��r�

and

���xh�� � ���x

h�Rh �

�Z�

��g�

�� �

����xh�Rh � �

����xh�� �

� ���xh�Rh

��d�����xh�� � ���xh�Rh

��

Variational Methods ��

As a result we get

���xh�� � ���x

h�Rh � �h

�Z�

��g�

�� �

����xh�Rh � ���h��Rh

���h��Rhd�������

�����������������������������

U�xh� � � �h�

�Rhinf

v��V�

� Z�

B��Rh

jvh�y��

�� �Z�

��g�

�� �����xh�Rh � ���h��Rh�d���

h��Rhy � v�jr�dy��r�

�� �h W h� �Rh��

�����

Now� if we set

W h�t� � Wh�t� � W h�t��

we arrive at the formula

U�xh� � � �hWh� �Rh�������

We remark that

��h�� � �������

W h��� �� Z�B

j�hjrdy��r

�� Z�B

jvhjr�dy��r�

������

From the above relations it follows that

W h��� �R�h ��h � �������

W h�t� �c��t�����

and ���� reads after scaling

ZBs

jr�hj�dy � c����� s��

nW h��� �Rh��h

o������

�� FUCHS et al�

Consider � � C���B� such that � � � � � in B and let ��x� � ��x�x

h

Rh��

Using ����� with � from above we �nd that

�����������

ZB

n� vh � div��� � � ���xh�Rh � �h�

h�� ��

�h����x

h�Rh �

� � � ���xh�Rh � �h�h�� g�� �� � g�����xh�Rh � �h�

h��ody � �

��� �

for all � � C��B� M n�nsym � and � � C�

��B� such that j �Dj � p�k� and � � � � ��

Let us �x an arbitrary number m �� Then� for all h large enough� theinequality

�� �hm �p�k� � �

�� �� �

p�k�

is valid�

Let

Km �n� � C��B� M n�n

sym � � j�Dj � mo�

and observe that the tensor � � ���xh�Rh��h� satis�es j �Dj � ���hm � �� �p�k� for any � in Km� Hence we can use inequality ��� �� Taylor�s formula

then implies

���������������������

ZB

n� vh � div���� � �h��� �

� �Z�

���g�

�� �����xh�Rh � ���h���

��

�d�� ��Z

���g��� �

����xh�Rh � ���h�h��h

�� �hd��

�ody � �

�����

for all � � Km and � � C�� �B� such that � � � � ��

By ����� and ����� we have �at least for a subsequence�

�h � $� in Lr�B� M n�nsym �������

Variational Methods ��

vh � $v in Lr��B�Rn�������

Now let us take arbitrary functions � � Km and � � C���B� with the restric�

tion � � � � �� Next we choose s � �t� �� so that spt� � Bs�

By ���� and ����� we can arrange �at least for another subsequence�

�h $� in Lr�Bs� Mn�nsym �������

�h $� a�e� in Bs������

Further we have

�ZB

jdiv�hjrdy��r

�� ZBRh�x

h�

jdiv�jrdx��r Rh

�hRnrh

�R��n�rh

�h

� ZBRh �x

h�

jf jrdx��r � Rh

�hsupx���

jf j

� Rh�hc���kfkW �

n���Rn�� � � as h ��

�see ������� So we have shown

div�h div$� � � in Lr�B�Rn�������

The relations ����� � ����� allow us to take the limit in inequality ����� forgiven � and � � In particular� from ������ ������ ����� we derive

ZBs

vh � div���� � �h��dyZBs

$v � div���� � $���dy������

For the last relation we took into account that

div$� � � in B�����

By ����� we �nd a tensor �� whose trace might be taken to be zero such that

��D�xh�Rh ��D and j��Dj � ��

�� FUCHS et al�

We have

�����������

���g����

����xh�Rh � ���h����� �

����g����

���D � ���h�D��

�� � �

���g����

����$��� $�

����

and using the same arguments

���g��� �

����xh�Rh � ���h�h��h

�� �h

���g��� �

����$��� $���� �

a�e� in Bs � ��� ���

Since the quadratic form � ���g����

������� � is non�negative� we conclude

with the help of Fatou�s lemma that

�������������������

lim infh��

ZB

�Z�

���g��� �

����xh�Rh � ���h�h��h

�� �hd��dy

�ZB

����g��� �

����$��� $�dy�

�����

From ���� it follows also that

���g����

����xh�Rh � ���h����� � �

�n�K�

tr�� � E����D�xh�Rh � ���h�D� �� �� �

� �n�K�

tr�� � c��g������j��D�xh�Rh � ��h�

Dj�j�Dj� �

� �n�K�

tr�� � c� sup���s��

�g������s�j�Dj� �

� c� ���� n�K�� c��j� j��

Variational Methods ��

We recall to the reader that� as it was shown in the proof of Lemma ������c� � �

Hk��

But then� by Lebesgue�s theorem� ���� and the above estimates

�����������������

limh��

ZB

�Z�

���g��� �

����xh�Rh � ���h����� �d��dy �

�ZB

����g��� �

������� �dy�

�����

Now from ������ ������ ����� and ����� we derive

���������

ZBs

n� $v � div���� � $���

���

���

�g����

������ � � � ���g����

����$�� � $��ody � �

�����

for all � � Km�

By arbitrariness of m it follows that inequality ����� holds for all� � C��B� M n�n

sym �� We know that $� � Lr�B� M n�nsym � and div$� � Lr�B� Rn�� It

is clear that� for any � � Lr�B� M n�nsym � with the property div� � Lr�B� Rn��

there exists a sequence �m � C��B� M n�nsym � such that

�m � in Lr�B� M n�nsym � and div �m div � in Lr�B�Rn��

But then it follows from ����� that

ZBs

h� $v � div����� ��

��g�

�� �����$�� � �

idy � � �� � C��B� M n�n

sym ��

Since the function � � C�� �B� is arbitrary �satisfying the condition � � � � ���

we �nally get

��$v� ���g�

�� �����$� in B�

Taking into account identity ����� we conclude that the pair $v� $� satis�es allconditions of Lemma ������ According to this lemma we have estimate �����

�� FUCHS et al�

in which the functions v and � have to be replaced by the functions $v and $� �respectively� Now� if we can show that

vh $v in Lr��Bt�Rn�������

�h $� in Lr�Bt� Mn�nsym �������

then the proof of the lemma is complete� Indeed� in this case� by ����� wehave

W �t� � �c��t�

and the lower semicontinuity of the norms with respect to weak convergenceimplies

W ��� � ��

But then

W �t� � c��tW ��� � c��t � W �t����

which is a contradiction�

We are now going to prove ����� and ������ ����� follows from ����� sinces � t�

Let � � C�� �Bt� M

n�nsym �� j� j � �� and consider a cut�o� function � such that

� � � in Bt� spt� � Bs� t � s � �� We insert these functions into ����� andarrive at the relation

�ZBt

vh � div� dy � �ZBs

�h � �vh �r��dy

�ZBs

�Z�

���g��� �

����xh�Rh � ���h�����

�d��dy �ZBs

j�hj jr�j jvhjdy � c�

ZBs

j� j�dy�

Variational Methods �

Using j� j � � we get for h su�ciently large

j��D�xh�Rh � �h���Dj � �� �h �

p�k� � �

�� �� �

p�k��

From this we obtainZBt

j��vh�j � sup��C�

��Bt�M

n�nsym ��

j� j��

��ZBt

vh � div�dy��

� c�� n�Z

B

j�hjrdy��r�Z

B

jvhjr�dy��r�

� �o�

By ������ ����� the sequence vh is bounded in BD�Bt�Rn� and hence pre�

compact in Lr��Bt�Rn� �see Appendix A��� This implies ����� and Lemma

���� is established��

We are now going to iterate Lemma ����� First observe the inequalities

���������j��D�x��tkR � ��D�x��Rj �

k��Pi��j��D�x��ti��R � ��D�x��tiRj�

j��D�x��ti��R � ��D�x��tiRj � t�nrU�x�� t

iR� � t�nr U�x�� t

iR��

�����

from which it is easy to get the estimates

�������������j��D�x��Rj � t�

nr

k��Pi��

U�x�� tiR� � j��D�x��tkRj �

j��D�x��Rj� t�nr

k��Pi��

U�x�� tiR��

�����

LEMMA ����� Let the numbers � ���� ��� t� ���� ��� t ���� t��� � ����p�k��

be chosen according to

�c���t�� ��t�� � �����

and dene �� and R� as in Lemma ����� Suppose further that the followingconditions hold

� � R � R�� BR�x�� �� �������

� FUCHS et al�

j��D�x��Rj � �� � � �p�k����� �

U�x�� R� �R� � �� � ��� t�� �minf��� ��� t �tnr ��� ���g�����

where �� ���� ��� Then� for any k � N� we have the estimates

j��D�x��tkRj � �������

U�x�� tkR� � t k�U�x�� R� �R�

�� t��� �k

�� t�� ������

Remark� Here we use the same notation as in Lemma �����

Proof� We will prove the lemma by induction� Let k � �� Then inequality���� follows from Lemma ����� In view of ������ ��� �� ����� we have

j��D�x��tRj � j��D�x��Rj� t�nr U�x�� R� � ���

�t�

nr �U�x�� R� �R��

�� t�� � �� �

t�nr ��

�� t�� �

� �� �t�

nr

�� t�� ��� t �t

nr ��� ������ t�� ��

Therefore estimate ����� for k � � is valid�

Suppose that all claims of the lemma are valid for k � �� �� � � � � s� We wish toprove them for k � s� �� Since

j��D�x��tkRj � ������

U�x�� tkR� � t k

�U�x�� R� �R�

�� t��� �k

�� t�� � � ��� k � �� � � � � s�����

we may use Lemma ���� replacing R there by tsR which gives

U�x�� ts�R� � t

�U�x�� t

sR� � �tsR����

t �s���U�x�� R� �R�

�� t��� �k

�� t�� �R�t��� �k

��

� t �s���U�x�� R� �R�

�� t��� ��s��

�� t�� ��

Variational Methods �

So ���� for k � s� � is proved� Moreover

j��D�x��ts��Rj � j��D�x��Rj� t�nr

sPk��

U�x�� tkR�

� �� �t�

nr

�� t�� �U�x�� R� �R�

� sXk��

t k

� �� �t�

nr ��

��� t�� ���� t �� �

which completes the proof��

LEMMA ����� Suppose that all the conditions of Lemma ���� are satised�Then we have the following estimates�

U�x�� � � ���

t�nr �

R� �U�x�� R� �R����� t�� ��������

j��D�x���j � �� � � � R� BR�x�� �� �������

Proof� Let ���� R�� We take k so that

tk� � �R � tk � R

tk �

t�

Then we have

U�x�� � �� Z�

B��x��

j� � ���x���jrdx��r � � Z

�B��x��

j� � ���x��tkRjrdx��r

�j���x��� � ���x��tkRj � �� Z�

B��x��

j� � ���x��tkRjrdx��r

� ��tkR�

�n� Z�

BtkR

�x��

j� � ���x��tkRjrdx��r �

� �� tkR��nrU�x�� t

kR��

�� FUCHS et al�

hence

U�x�� � � ���t�nr U�x�� t

kR� � ���t�nr t k�

U�xo� R� �R�

�� t�� � ��

t�nr ��

t

R� U�xo� R� �R�

�� t�� �

and ���� follows� Next we observe

j��D�x��� � ��D�x��Rj � j��D�x��� � ��D�x��tkRj�

�k��Pi��j��D�x��ti��R � ��D�x��tiRj �

� ��t�nr U�x�� t

kR� � ��t�nr

k��Pi��

U�x�� tiR� �

� ��t�nr

kPi��

U�x�� tiR� � ��

t�nr

�� t�� U�x�� R� �R�

�� t �

���t

�nr

��� t ���� t�� �� t�

nr ��� t�� ���� t �t

nr ��� ���

��� t ���� t�� �

� �� ���

This inequality implies ������

Variational Methods ��

��� Proof of Theorem �����

As in the proof of the Main Lemma ���� we are going to consider the saddlepoint �u� �� obtained in Lemma ������

We �rst deduce from inequality ���� the relation

ZBR�x��

n� uR � div���� � ��� � ���R � �� � ��� g���� � g�����

odx � �����

being valid for all � � C��BR�x��� Mn�nsym � and � � C�

��BR�x��� such that

j�Dj � p�k� and � � � � �� Here

uR � u� �R�x� x��� v�R� v�R � V�� �R ��g�

������x��R�

and it is assumed that

j��D�x��Rj � � �p�k������

We now use the same arguments which led to ��� � and ������ We write ����in the form

ZBR�x��

n� uR � div���� � ���x��R � ���x��R � ��� � �

��R � �� � ���x��R�

�g��� � ���x��R � ���x��R�� �R � �� � ���x��R�

�g��� � ���x��R � ���x��R��odx � ��

and then� by applying Lagrange�s formula� estimates ����� ���� and also im�posing the additional restriction

j�D � ��D�x��Rj �p�k� � �

������

��� FUCHS et al�

we have

�������������������������������������������������������������������������������

� �Z

BR�x��

n� �uR � div�� � ���x��R�� �uR �r�� � �� � ���x��R�

��uR � div�� � ���x��R� � �uR �r�� � �� � ���x��R�

���

D �Z�

���g��� �

����x��R � ���� � ���x��R���� � ���x��R���

�� � ���x��R�d�� ��Z

���g��� �

����x��R � ���� � ���x��R���� � ���x��R���

�� � ���x��R�d��Eo

dx

�Z

BR�x��

n� �uR � div�� � ���x��R�� �uR �r�� � �� � ���x��R�

��uR � f � juRjjr�jj� � ���x��Rj

���

htr�������x��R�

n�K�� sup

jsj�p�k����

�g������s�j�D � ��D�x��Rj�

iodx�

����

For the second inequality we have used ���� and the following elementaryestimate

j��D�x��R � ����D � ��D�x��R�j � j��D�x��Rj� j�D � ��D�x��Rj

�p�k��

Let the cut�o� function � satisfy � � � on BR��x��� jr�j � �Rin BR�x��� We

insert � � ���x��R�p�k���

� into ����� assuming that � � C���BR�x��� M

n�nsym ��

j�j � � on BR�x�� and spt� � BR��x��� Then the tensor � satis�es restriction���� and we can derive from ���� the following inequality

ZBR�x��

n� uR � div�

p�k� � �

�� �

p�k� � ���

�c���n�K�� �� k��j�j�

odx

�Z

BR�x��

juRj jf jdx� �

R

ZBR�x��

juRj j� � ���x��Rjdx

Variational Methods ���

being valid for all � � C�� �BR��x��� M

n�nsym � such that j�j � � in BR��x���

Here

c�� � maxn �

nK�

� maxjsj�

p�k����

�g������s�

o�

Choosing a rigid displacement v�R � V� so that

U�x�� R� ��

R

� Z�

BR�x��

juRjr�dx��r�

���

and recalling the de�nition of how to apply a convex function to a measure�we obtain Z

BR���x��

H�j��uR�j� �

sup��C�

��BR���x���M

n�nsym ��

j�j�� in BR���x��

ZBR���x��

n� uR � div � � c��

�p�k� � ��

�j�j�

odx

� �p�k��

ZBR�x��

juRj�jf j� �j� � ���x��Rj

R

�dx � jBR�jS�x�� R��

where

S�x�� R� � �n��p�k��

hR� Z�

BR�x��

jf jrdx��r

� �U�x�� R�i U�x�� R�

� S��x�� R� � �n�p�k��c��

�kfkW �

n���Rn�

�hR � U�x�� R�

i U�x�� R�

and

H�t� �

���t�

c���p�k��� if jtj �

p�k���

c��

jtj �p�k���

c�� if jtj p�k���

c��

� supnst�

p�k���

c��s� � jsj � �

o�

After application of Jensen�s inequality �see Appendix A��� we arrive at therelation

H� Z

�BR���x��

j��uR�j��

Z�

BR���x��

H�j��uR�j� � S��x�� R�����

��� FUCHS et al�

Using the imbedding theorem �see Appendix A��� it is easy to establish theestimate�������������

�R

� Z�

BR���x��

j uRjr�dx��r� � c���n� r�

Z�

BR���x��

j�� uR�j

� c��

Z�

BR���x��

j��uR�j��� �

where

uR � u� �R�x� x��� v�R� v�R � V��

and Z�

BR���x��

j uRjr�dx � infv��V�

Z�

BR���x��

ju� �R�x� x��� v�jr�dx�����

So we can replace ��� by

H

�B� �

c��R

� Z�

BR���x��

j uRjr�dx��r��CA � S��x�� R������

Let us consider two cases� Suppose �rst that

Rc��

� Z�

BR���x��

j uRjr�dx��r� � p

�k� � �

�c��������

Inequality ���� in this case takes the form

�p�k� � ��c��c���

� �R

Z�

BR���x��

j uRjr�dx��r� � S��x�� R��

From this we get�����������������������������

�R

� Z�

BR���x��

j urjr�dx��r� � q

�p�k� � ��c��c

���S��x�� R�

��

�n�p�k��c���R � U�x�� R�� U�x�� R��

p�k� � ��c��c

���

����qc��c��c����n��R � U�x�� R��

�� U���x�� R�

� �� U�x�� R� � c��c��c

����

n��R � U�x�� R���

�����

Variational Methods ���

In the opposite case we �rst note that

��

�c� R

�Z�BR���x��

j uRjr�dx��r� �

� H

Rc�

�Z�BR���x��

j uRjr�dx��r�� � S��x�� R��

and therefore

��������R

� Z�

BR���x��

j uRjr�dx��r� � �c��S��x�� R� �

� �n��p�k��c��c���R � U�x�� R�� U�x�� R��

�����

Putting together estimates ����� and ����� we get the �nal result

��������R

� Z�

BR���x��

j uRjr�dx��r� � max f�

�� c���R � U�x�� R��g U�x�� R�

�c���R � U�x�� R���

�����

where

c�� ��n�

p�k� � �

c��c��� c�� � c��c��c����

n��

Now� we can prove the following

LEMMA ����� Let BR�x�� b �� and suppose that

j��D�x��Rj � �� �nrU�x�� R� �� � � �

p�k��������

Then we have the estimate

U�x�� R��� � max f��� c���R�U�x�� R��g U�x�� R��c���R�U�x�� R�������

where the constant c�� depends only on K�� n� �� r� k�� kfkW �n���Rn��

��� FUCHS et al�

Proof� We will use estimate ������ Taking into account formula ��� inwhich R is replaced by R�� �see also ������ we have

U�x�� R��� ��R

� Z�

BR���x��

ju� �R��x� x��� v�R�jr�dx��r�

� �R

� Z�

BR���x��

ju� �R��x� x��� v�Rjr�dx��r�

� �R

� Z�

BR���x��

j uRjr���r�

� j�R� � �Rj

� maxn��� c���R � U�x�� R��

o U�x�� R� � c���R � U�x�� R�� � j�R� � �Rj�

On the other hand� we also have

j���x��R� � ���x��Rj � �nrU�x�� R������

Due to ����� and ���� we get

j��D�x��R � ����D�x��R� � ��D�x��R�j � ���� �

for any � � ��� ��� Lagrange�s formula gives

j�R� � �Rj � j�g���

����x��R��� �g���

����x��R�j

��Z

j��g�

�� �����x��R � �����x��R� � ���x��R������x��R� � ���x��R�j d�

� maxn

�nK�

� supjsj�

�g������s�

oj���x��R� � ���x��Rj

� �nr max

n�

nK�� supjsj�

�g������s�

oU�x�� R��

If we put

c�� � c�� � �nr max

n �

nK�� supjsj�

�g������s�

o�

then we arrive at ����� Lemma ���� is proved��

Variational Methods ��

LEMMA ����� Let x� � �� be such that

limR��

U�x�� R� � �������

limR��

j��D�x��Rj � �� �p�k�������

Then

lim infR��

U�x�� R� � �������

Proof� By conditions ����� and ����� there is a number R� � such that

j��D�x��Rj �p�k��

�and �

nr U�x�� R� �

p�k���

�for all R ���� R��� Therefore

we have

j��D�x��Rj ��p�k� � ���

� �nr U�x�� R������

for all R ���� R���

According to ����� there exists a number R� � having the property

c���n�K�� r� kfkW �

n���Rn��

�p�k� � ���

�U�x�� R� �

������

for any R ���� R��� But then� by ������ ����� and Lemma ����� we have

����� U�x�� R��� � �

� U�x�� R� � c��

�n�K�� r� kfkW �

n���Rn��

�p�k���

��R � U�x�� R�� for any R ����minfR�� R�g��

�����

In ����� and ����� we used the following notation� the constants c�� and c��de�ned before depend on a parameter � and we now replace � by the quantity�p�k���

Iteration of ����� gives

U�x�� R��k� � �

�k U�x�� R� � c��

k��Xi��

�k���i�U�x��

R

�i� �

R

�i

������

�� FUCHS et al�

for any R ����minfR�� R�g��

It is known that

limk��

k��Pi��

�i�U�x��

R�i� � R

�i

��k��

� limk��

�k�U�x��

R�k� � R

�k

��k � �k��

provided the limit of the right hand side exists� But by ����� this limit isequal to zero� Now� it follows from ����� that

U�x��R

�k� � as k ���

Lemma ���� is proved��

Proof of Theorem ������ Let us choose a number r ��n� �nn�� � such that

r � n and introduce the sets

�� �nx� � � � lim sup

R��U�x�� R� �

o�

�� �nx� � � � � lim

R��j��D�x��Rj

o�

It is clear that j�j � � where � � �� � ��� Let us represent the set � n � asthe union of the two disjoint subsets

�� � fx� � � n � � j�D�x��j �p�k�g

�� � fx� � � n � � j�D�x��j �p�k�g�

We are going to show that the set �� is open and � � C ���� Mn�nsym � for any

� ���� ���

To begin with� we take an arbitrary number � ���� �� and de�ne

t� � ������

��� � ��

Let x� � ��� We choose some domain �� such that x� � �� �� � and let

� �j�D�x��j�

p�k�

��

Variational Methods ��

Now we can determine the constant c���t�� �� from Lemma ������ Without lossof generality let c�� be greater then �%��

If we set

t � ����c����

��� �

then t ���� t�� and identity ���� holds� So we can determine the constants ��and R� of Lemma ���� by choosing �� �

� ��

By Lemma �����

lim infR��

U�x�� R�

and since

limR��

j��D�x��Rj � �� �j�D�x��j� �

�� � �

p�k��

a number R � exists such that conditions ���������� hold�

The functions x j��D�x�Rj and x U�x�R� are upper semicontinuous atthe point x�� and so a nonempty ball Br��x�� exists such that at each point ofBr��x�� conditions ���������� are satis�ed� i�e�

BR�x� �� �� �� � R � R��� j��D�x�Rj � ���

U�x�R� �R� � �� for any x � Br��x���

But then� by Lemma ����� we have the estimates

U�x� � � ���t�nr � �

R� U�x�R� �R�

�� t�� �

� ���t�nr ��

�� t�� �

R� � � � � R� x � Br��x���

It follows that the function x ��x� is H&older continuous with exponent � insome neighborhood of the point x�� and moreover� all points of this neighbor�hood belong to ��� This means that �� is an open set and � � C ���� M

n�nsym ��

Theorem ����� is proved��

��� FUCHS et al�

��� Open Problems

The following problems should be investigated�

a� To prove the statement of Theorem ����� by direct methods� and so to getexplicit integral conditions under which a given point of an elastoplastic bodybelongs to the elastic or plastic zone� Partially this problem has been solvedin �Se���� In the twodimensional case it turned out to be possible to apply theso�called �hole��lling trick� and to get the following results� we introduce thefunction

!�x�� R� �Z

BR�x��

jr�j�dx

and the sets

�� � fx� � � � limR��

j��D�x��Rj �p�k�g

�� �nx� � � � lim sup

R��

!���x�� R� ln�R

ln� ���x��R�R� �

�o�

THEOREM ���� Suppose that n � � and let conditions������� ������ ofTheorem ����� and condition ������ hold� Assume further that the functionx �D�f�x�� is continuous in �� Then the set �� � �n�� where � � ������is open� Moreover� the tensor � is continuous in �� and

F���x�� � � for all x � ���

It is easy to see that the twodimensional Lebesgue measure of the set �� isequal to zero� i�e� j��j � �� However� using a ��measure of Hausdor�� onecan establish more precise estimates for the set ��� Suppose that � is somenondecreasing positive function on the segment ��� ��� E is an arbitrary set inRn � We let

H�E��� � lim inf���

fXi

��ri� � E � �i

Bi� ri � �g�

Here fBig is an arbitrary countable cover of the set E by open balls Bi ofradius ri � �� Applying known arguments �see� for example� �Gi��� one can

Variational Methods ���

show that H���� �� � � for the choice ��t� � �ln �t����

If we writeHm�E� in place ofH�E��� for the choice ��t� � tm� thenHm���� �� for any � � m � �� and therefore the Hausdor� dimension of �� is equal tozero�

b� To prove a global variant of Theorem ������ i�e� to show that� � W �

� ��� Mn�nsym ��

c� To prove that

� � W ���loc��� M

n�nsym ��

We note that in the case of elastoplastic torsion this result is correct �see� forexample� �BS�� �CR�� �F���

d� To study the properties of the internal free boundary dividing an elastopla�stic body into elastic and plastic zones�

��� FUCHS et al�

�� Remarks on the regularity of minimizers

of variational functionals from the defor�

mation theory of plasticity with power

hardening

In this section we consider the variational functional from ������ i�e� we let

I�v� �Z�

g���v�� dx�M�v��

g��� � ��K�tr

�� � g��j�Dj�� � � M n�n

sym �

with a function g� � R R having superlinear growth� More precisely� weassume that the following conditions hold�

�������������������������������������������

�i� g� is of class C�� g���� � �� g����� � ��

�ii� the function t g���t� is continuously di�erentiablefor all t except perhaps at � t�� The left and rightderivatives �g������t�� and �g�����t�� exist� The functiont g����t� is extended to t� by means of the value�g�����t���

�iii� There are positive numbers c� and c� such that

c��� � t������ � h��t� �� minfg����t�� g���t��tg �

� h�t� �� maxfg����t�� g���t��tg �� c��� � t��

���� � t � ��

�����

for some

� � � � �������

We remark that in the deformation theory of plasticity based on the von Misesplasticity test� the range of admissible values for � corresponds to the model ofan elasto�plastic body with power hardening� The case � � � describes linearhardening�

The natural domain of the functional I is the Banach space

D���� � fv � div v � L����� jvj� j�D�v�j � L�����g�

Variational Methods ���

If we assume that we are given functions

f � Ln���Rn�� F � L������Rn�� u� � W �

� ���Rn�������

then the variational problem

��� to �nd u � V� � u� � fv � D������� � v � � on ���g � u�

such that I�u� � inffI�v� � v � V� � u�g�����

has a unique solution�

This follows from the fact the the strictly convex functional I is continuous onD������� and coercive on the a�ne submanifold V��u� of the re�exive spaceD��������

We wish to describe brie�y what regularity results for minimizers of ����� canbe expected� Proofs of the corresponding statements can be given via dualityapproach and may be obtained by nonessential modi�cations of the proofs ofTheorem ����� and ������ For details we refer the reader to the paper �Se ��

Now let us formulate the dual problem of �����

��� to �nd � � Qf such that

R��� � supfR��� � � � Qfg�����

where

Qf �� f� � � �Z�

� � ��v�dx � M�v� �v � V�g

is the set of all tensor �elds satisfying the equilibrium equations in terms ofstresses�

� �� f� � L���� M n�nsym � � j�Dj � L

���� ���g

is the space of admissible tensors�

R��� ��Z���

���� � u�d�Z�

g����dx

��� FUCHS et al�

is the functional of the problem�

Z���

���� � u�d ��Z�

� � ��u��dx�M�u���

g���� ��

�n�K�

tr�� � g���j�Dj��

and g���s� � supfst� g��t� � t � Rg is the conjugate function of g��

It is well known that problem ���� also has a unique solution which is connec�ted to that of ����� by the relations

���I�u� � R����

� � K� div u ��g���j�D�u�j�j�D�u�j �D�u��

����

We assume that the following density condition holds�

V� � V� �W �� ���R

n� is dense in V� w�r�t� the norm of D����������� �

and refer the reader to Appendix A�� concerning tests for ��� ��

The main results of this section� announced in �Se� and proved in �Se �� aregathered in the following assertions�

THEOREM ����� If conditions ����� ����� and ����� hold and if further

f � W ��������Rn�������

then

� � W ���loc��� M

n�nsym ��

Variational Methods ���

THEOREM ����� If conditions ����� ����� and ����� hold and if

f � W �n���R

n�������

for some

n n�� � ��

n� � � � ��������

then there exist open sets �� and �� in � and a number �� � ���n� n� ���� ��such that

� � C ��� � ��� Mn�nsym �

for all � ���� ��� and���������������

F���x�� �� j�D�x�j � p�k� � � for all x � ��

F���x�� � for all x � ��

F���x�� � � for almost all x � � n ��� � ���

wherep�k� � g���t���

Under conditions ����� and ����� the dual function g�� coincides with the Le�gendre transform of g�� i�e�

g���s� � st� g��t�� s � g���t��

Consequently� g�� is a function of class C�� in addition� its second derivativeexists and satis�es

�g������s� � �

g��� �t�� s � g���t�� t � �g���

��s��

��g�������

p�k�� � �

�g��� ���t���

From the second identity in ���� we obtain

��u� ��

n�K�tr � ��

�g�����j�Dj�j�Dj �D�

��� FUCHS et al�

Hence� we deduce that

��u� � C ��� � ��� Mn�nsym �

for every � ���� ���� The di�erentiability properties of the minimizer of problem����� on �� � �� are further improved by means of a well�known scheme andare determined only by the smoothness of the data g� and f �

REMARK ����� Conditions ������ ����� and ������ are not optimal� In par�

ticular� the claim of Theorem ����� is true if f � L����

loc ���Rn��

REMARK ����� In a number of practically meaningful cases the sets ��

and �� may be interpreted as the elastic and plastic domains�

We comment brie�y on the above remark� Let g��t� � �t� for t � ��� t��where � is a positive constant� Then� from Theorem ������ it follows that theequilibrium equations of linear elasticity are satis�ed in ��� i�e�Z

��

fK� div u div v � ���D�u� � �D�v�� f � vgdx � � �v � C�� ����R

n��

and that the deformation tensor is computed according to the formula for anelastic material

��u� ��

n�K�tr � ��

���D�

Hence it is natural to call �� the elastic domain�

On the open set �� we have the quasilinear system of di�erential equationsdescribing the equilibrium of an elasto�plastic body

Z��

fK� div u div v �g���j�D�u�j�j�D�u�j �D�u� � �D�v�� f � vgdx � �

�v � C�� ����R

n��

and the plastic deformation

�p � ��u�� �e � ��u�� �

n�K�tr � �� �

���D �

��g�����j�Dj�j�Dj � �

��

��D

Variational Methods ��

is nonzero at every point of this set� Due to this facts �� may be called theplastic domain�

Unfortunately we have considerably less information on �� � � n ��� � ����We mention only that at almost all points the plastic deformation is zero andj�D�u�j � t�� Formally the set �� could be called the elasto�plastic boundary�since it separates the elastic and plastic domains� This set could again bepartitioned into subsets� each having its own nature� The elements of such apartition could be� for example� ���� ���� and int ��� The last set� generallyspeaking� is nonempty� even more� it is easy to give an example where thisset coincides with the entire domain �� Moreover� on the set int �� the equi�librium equations for a linear elastic medium and those for an elasto�plasticmedium are satis�ed simultaneously� since they coincide there� Finally� ��may contain singular points� where a loss of smoothness may occur since thevariational problem ����� is considered in a class of vector�valued functions�

THEOREM ����� If the conditions of Theorem ����� hold and� in addition�

g����t� is continuous at t��

then there exist an open set � � � and a number �� � ���n� n� ���� �� suchthat

� � C � �� M n�nsym �

for all � ���� ��� and j� n �j � ��

REMARK ����� From Theorem ����� and the second relation in ����� itfollows that the solution of problem ����� is regular on an open set of fullmeasure if the data g� and f are su�ciently regular�

At the end of this section we remark that if n � �� then ��u� and � are con�tinuous functions in �� The proof of this fact for a more di�cult case will begiven in Chapter ��

Appendix A

A�� Density of smooth functions in spaces of

tensor�valued functions

In what follows we assume that � is a bounded domain in Rn whose boundaryis Lipschitz continuous�

We denote by MN�n the space of all real N � n matrices� Adopting the

convention of summation over repeated Latin and Greek indices running from� to N and from � to n� respectively� we will use the notation

� � � � �i��i�� � � ��i��� � � ��i�� � M N�n � j�j � p� � ��

LEMMA A���� Let � � L���� M N�n�� div � � ��i���� � Ls���RN � and let� � s � t � ��� Then a sequence �m � C���� M N�n� exists such that

�m � in Lt��� M N�n�������

div �m div � in Ls���RN �������

�m�� � in L���� M N�n�������

k�mkL���� � k�kL����������

Proof� Since the boundary of � is Lipschitz continuous� for any pointx � �� there is a neighborhood Ox such that the domain Ox � � is star�shaped relative to some of its points� All these neighborhoods together with

��

Variational Methods ��

� form an open cover of the compact set �� and so a partition of unity existssuch that�����������������

�k � C�� �Rn�� �k � � in Rn � k � �� �� �� � � � � r�

spt �� � � � ��� spt �k � �k � Oxk � �� k � �� �� � � � � r�

rPk��

�k � � in ��

We set �k � �k�� k � �� �� � � � � r� Let x�k � �k be a point with respect to which�k is star�shaped� Now we stretch the domain �k with the help of a similaritytransformation relative to the point x�k� In the domains

�k � fx � R

n � x�k � ��x� x�k� � �kg� � � � � ��

we de�ne the functions

�k �x� � �k�x�k � ��x� x�k��� x � �k

Further we let

���k �x� �Z�k�

���x� y��k�y�dy� ����x� �Z�

���x� y����y�dy�

where �� is the standard smoothing kernel� i�e�

��x� � cn

�������e

jxj��� if jxj � �

� if jxj � �

�ZRn

��x�dx � ��

a positive parameter and ���z� � �n��z� ��

Under the conditions of the lemma div �k � Ls���RN �� and� using knownarguments �see� for example� �So��� �So���� we get that� for any � �� anumber ���� exists such that� for any � ���� ����� ��� there is a number ���� ��such that���������

k��� � �kLt��MN�n � � �� kdiv ���� � ��kLs��RN � � ��

���� ���� ���� ��� � ��� �rP

k�����k �

��� FUCHS et al�

For j����x�j we have the estimate �R � k�kL��

j����x�j � Rh Z�

���x� y����y�dy �rX

k��

Z�k�

���x� y��k�y�dyi�

Decreasing if necessary the numbers ���� and ���� �� we can arrange that�� � �� � � � �� in conclusion

k���kL���� � �� � ��R�

Now we take � � �m� �m � �� �

��� �

m�� m � �

����m�

�m� and put �m � �m��m�

Then we obtain the statements ����������� of Lemma A���� together with theinequality

lim supm��

k�mk � R�

On the other hand

k�mkL���� � supk�kL������

Z�

�m � �dx�

From this we conclude that

lim infm�� k�

mkL���� � R � lim supm��

k�mkL�����

which implies

R � limm�� k�

mkL�����

Then the sequence �m � �m�m� where �m � R�k�mkL����� satis�es all the

requirements� Lemma A���� is proved��

If we proceed in the same way� we get the following statements�

LEMMA A���� Let � � L���� M N�n� and � � t � ��� Then a sequence�m � C���� M N�n� exists such that ������ ������ ����� hold�

Variational Methods ���

LEMMA A���� Let � � Lt��� M n�nsym �� �D � L���� M n�n

sym �� div � � Ls���Rn�

and � � s � t � ��� Then a sequence �m � C���� M n�nsym � exists which has

the properties

�m � in Lt��� M n�nsym ������

div �m div � in Ls���Rn������

�mD �� �D in L���� M n�n

sym � ���� �

k�mDkL���� � k�DkL����������

LEMMA A���� Suppose that � � Lt��� M n�nsym �� �D � L���� M n�n

sym � and � �t � ��� Then a sequence �m � C���� M n�n

sym � exists such that the claims������ ����� and ����� hold�

LEMMA A���� Suppose that � � �t � f� � � � L���� M n�nsym �� �D �

L���� M n�nsym �� div � � Lt���Rn�g for some � � t � ��� Then � � Lt��� M n�n

sym ��

Proof� We have the identity

Z�

� � ��v�dx � �Z�

v � div � dx �v � C�� ���Rn��

From this it follows that

n

Z�

tr � div v dx � �Z�

��D � ��v� � v � div ��dx �v � C�� ���Rn�������

We denote by�Dq the closure of C

�� ���Rn� with respect to the norm krukLq����

by Hq the closure of all solenoidal vector�valued functions from C�� ���Rn�

with respect to the same norm and by $Hq the space of solenoidal vector�valued

functions from�Dq� We set q � t

t�� ���� ��� Since � is a bounded Lipschitz

domain� we have Hq � $Hq �see for example �P���Next� by �D � L���� M n�n

sym �� the linear functional

v g�v� �Z�

��D � ��v� � v � div ��dx

��� FUCHS et al�

is bounded on�Dq� It follows from ����� that it is equal to zero on Hq � $Hq�

Therefore �see for example �P��� there exists p � Lt��� such that

g�v� �Z�

p div v dx�

So we have the identity

Z�

��

ntr � � p�div v dx � � �v � �

D� �

Since the range of the operator G ��D� L����� Gu � div u� is the set of all

functions from L���� which are orthogonal to �� we get �ntr� � p � constant

�see for example �LS��� Lemma A��� is proved��

A�� Density of smooth functions in spaces of

vector�valued functions

Let us consider the space of vector�valued functions � Rn de�ned as

Dp�q��� � fv � �vi� � kvkDp�q��� �� kdiv vkLp��� � kvkLq��� � k�D�v�kLq��� � ��g�

where p� q � � and � denotes a bounded Lipschitz domain� We suppose that

� � q � p � ��� n � q������

From this it follows that

Dp�q��� � Dq�q����

and we have

D������ is continuously imbedded into Ln

n�� ���Rn�������

Dq�q��� � W �q ���R

n� if q �������

Variational Methods ���

The reader can �nd a proof of statement ����� for instance in �MM��� �����follows from Lp�Korn�s inequality �see �MM��� �MM���� we also refer to Lemma����� in chapter ��

By the imbedding theorem �for Sobolev spaces� we get �q �������the space Dq�q��� is imbedded continuously into theLebesgue space Ls���Rn� for s � nq

n�q � if n q�

and for any s � ��� if n � q�

�����

We impose the additional restriction

p � nq

n� qif n q and p � �� if n � q�����

It follows from ����� that if p satis�es conditions ����� and ����� then we candeduce

�v � Dp�q��� for all � � C���� and all v � Dp�q��������

To prove this it is enough to look at the formula

div ��v� � � div v �r� � v � Lp��� for � � C����� v � Dp�q����

With standard arguments �see Lemma A���� and also �So��� �So��� and remem�bering the fact that ���� implies ����� we get

LEMMA A���� Suppose that condition ����� holds� Then

C����Rn� is dense in Dp�q������� �

It is known �see �MM��� that

Dp����� is imbedded continuously in L�����Rn�������

and� for q �� by ����� and the imbedding theorem for Sobolev spaces

Dp�q��� is imbedded continuously in Lq����Rn�������

Therefore we can de�ne the space

Dp�q� ��� � fv � Dp�q��� � v � � on ��g�

Exploiting the same ideas as in the proof of Lemma A����� it can be proved�

��� FUCHS et al�

LEMMA A���� If condition ����� holds� then

Dp�q� ��� � Dp�q

� ����������

where the space Dp�q� ��� is dened as the closure of C�

� ���Rn� with respect tothe norm of the space Dp�q����

Next� let ���� ��� � �� be such that

��� � ��� � �� ��� � ��� � ���

We introduce the subspace

V p�q� ��� � fv � Dp�q��� � v � � on ���g�

We are interested under what conditions concerning ��� and ��� the followingstatement is correct

V p�q� ��� � C����Rn� is dense in V p�q

� ����������

Lemma A���� and A���� show that ������ holds if condition ���� and ��� � �

or ��� � � are satis�ed�

There is another interesting case in which ������ holds�

LEMMA A���� Suppose that condition ����� is fullled and that there arefunctions �� and �� from C�

� �Rn� such that �� � �� � � in �� �� vanishesin some neighborhood of ��� and �� vanishes in some neighborhood of ����Then ������ is true�

The statement of Lemma A���� directly follows from Lemma A���� and LemmaA�����

One may also ask the question whether statement ������ is valid if condition���� does not hold� The following lemma shows that statement ������ is correctfor any p satisfying condition ����� provided it is correct for some p satisfyingcondition �����

Variational Methods ���

LEMMA A���� Suppose that the following conditions are satised�

q � n�nq

n� q� p � ���������

V r�q� ��� � C����Rn� is dense in V r�q

� for some r � �q� nqn�q ��

if q �� and for some r ���� nn�� �� if q � ��

������

Then we have statement �������

Proof� We consider the spaces

Y � �� Lq�����Rn�� Y � � �� � Lq������R

n�

where ���������������

� � f� � tr� � Lp���� �D � Lq��� M n�nsym �g�

�� � f�� � tr�� � Lp����� ��D � Lq���� M n�nsym �g�

p� � pp�� � q� � q

q�� �

The duality relation between Y and Y � takes the form

hy�� yi �Z�

�� � �dx �Z���

b� � bdl� y � f�� bg� y� � f��� b�g�

We introduce the operator A � Dp�q��� Y �

Av � f��v���vj���g�

and note� by the imbeddings ����� and ������ there are constants c� and c� suchthat

c�kvkDp�q��� � kAvkY � c�kvkDp�q��� �v � V p�q� ����������

Let � �V p�q� ������ We will show that an element y� � Y � exists such that

�v� � hy�� Avi for any v � V p�q� ���� Indeed� it follows from ������ that

j�v�j � c�kvkDp�q��� � c�c�kAvkY �v � V p�q

� ���������

��� FUCHS et al�

Let us consider the subspace Y� � A�V p�q� ���� � Y and introduce on Y� the

linear functional g�y� � �A��y�� By �����

jg�y�j � c�c�kykY �y � Y��

Let G denote a norm preserving extension of g to the whole space Y � Thenthere exists y� � Y � such that G�y� � hy�� yi for all y � Y and g�y� � G�y�for y � Y�� We have

g�Av� � hy�� Avi � �v� �v � V p�q� ����

But it means that

�v� � �Z���

b� � vdl �Z�

�� � ��v�dx �v � V p�q� ���� y� � f��� b�g�

Now we start with the proof of the lemma� We suppose that the claim of thelemma is false� i�e�

�v� � V p�q� ��� n V p�q

o ��� � C����Rn�������

Then there exists � �V p�q� ����� such that

��� �v� � � �v � V p�qo ��� � C����Rn�

�v�� � ��

As it was proved above one can �nd �� � �� and b� � Lq������Rn� with the

properties

������������Z���

b� � v�dl �Z�

�� � ��v��dx � ��

�Z���

b� � vdl �Z�

�� � ��v�dx � � �v � V p�q� ��� � C����Rn��

���� �

It follows from ���� � that

n

Z�

tr��div v dx � �Z�

��D � ��v�dx �v � C�� ���Rn��

Variational Methods ��

We put

h�v� � �Z�

��D � ��v�dx

and note that h�v� � � for all smooth solenoidal vector �elds with compactsupport in �� Since �� � Lq���� M n�n

sym ��

jh�v�j � constantkvkW �q ��R

n��

The latter implies �see �P�� the existence of a function t � Lr����� where

r� �

��� q� if q �

r� � rr�� if q � ��

such thatZ�

tdiv v dx � �Z�

��D � ��v�dx � � �v � C�� ���Rn��

So we have

Z�

��

ntr �� � t�div v dx � � �v � C�

� ���Rn��

and therefore

ntr �� � t � constant in ��������

By the conditions of the lemma we know r� � rr�� � r�� and thus we can state

that

tr �� � Lr�����������

Since r � nqn�q � v� � V r�q

� ���� and in view of ������ a sequence vm � V r�q� ��� �

C����Rn� � V p�q� ��� � C����Rn� exists such that

vm v� in Dr�q����������

�� FUCHS et al�

It follows from ������ and ������ that

Z�

�� � ��vm�dxZ�

�� � ��v�dx�������

Recalling that r � q� we also have

vm v� in Lq�����Rn��

From this and ���� �� ������ we get the contradiction

� �Z�

�� � ��vm�dx�Z���

b� � vmdlZ�

�� � ��v�dx�Z���

b� � vdl � ��

Lemma A���� is proved��

A�� Some properties of the space BD��� IRn�

We recall that the space BD���Rn� is de�ned as the set of vector�valuedfunctions with �nite norm

kvkBD��Rn� � kvkL���� �Z�

j��v�j������

where Z�

j��v�j � supf�Z�

v � div � dx � � � C�� ��� M n�n

sym �� j� j � � in �g������

We note that in this case the strain tensor corresponding to the displacement�eld u generates a bounded positive Radon measure � which on open subsets� � � is given by

���� �Z�

j��v�j � supf�Z�

v � div � dx � � � C�� ��� M n�n

sym �� j� j � � in �g�

For more information concerning BD���Rn� we refer to the papers �AG�� and�ST��� we are just going to reprove some of its properties�

Variational Methods ��

THEOREM A���� ��AG��� �ST��� Suppose that � is a bounded Lipschitzdomain� Then���������

the space BD���Rn� is continuously imbedded into the space

Lp���Rn� for p � ��� nn�� �� moreover� for p � ��� n

n�� �

this imbedding is compact�

�����

�����for each r � ��� n

n�� � a constant C � C��� r� n� exists such that

infv��V�

� R�jv � v�jrdx

��r � CR�j��v�j �v � BD���Rn��

�����

We begin with

LEMMA A���� Let � be a locally star shaped domain� Then� for any u inBD���Rn�� a sequence fumg�m�� � C�

� �Rn �Rn� exists such that

um u in L����Rn�����

limm��

Z�

j��um�jdx �Z�

j��u�j�����

Proof� Local star�shapedness of � just means that for any point x � ���there is a neighborhood Ox such that Ox � � is star�shaped with respect tosome of its point x�� We have

� � � � ��

x���Ox��

and since � is compact� a �nite subcover exists such that

� �r�

k��

�k�

where �� � �� �k � Oxk � �� and �k is star�shaped with respect to x�k�k � �� �� � � � � r� Let f�kgrk�� � C�

� �Rn� be a partition of the unity correspon�ding to this subcover� i�e������������������

spt �k � �k� k � �� �� �� � � � � r�

� � �k � � in Rn � k � �� �� �� � � � � r�

rPk��

�k � � in ��

��� FUCHS et al�

Let � be a parameter from ��� ��� We introduce the sets

�k � fz � Rn � x�k � ��z � x�k� � �kg� k � �� �� � � � � r�

and the functions uk � �ku� �k � u�r�k� k � �� �� �� � � � � r�

�k�x� � �k�x�k � ��x� x�k��� x � �

k � k � �� �� � � � � r�

uk�x� � uk�x�k � ��x� x�k��� x � �

k � k � �� �� � � � � r�

�k �x� � �k�x

�k � ��x� x�k��� x � �

k � k � �� �� � � � � r�

For all � close to � we have

� � spt�k � �k� k � �� �� � � � � r���� �

Thus we may extend all functions uk� �k by zero to � and obtain�������

kuk � ukkL���k� � kuk � ukkL���� ��� ��

k�k � �kkL���k� � k�

k � �kkL���� ��� �� k � �� �� � � � � r�

�����

We further let�����������������

�u����x� �R����x� y�u��y� dy� ������x� �

R����x� y����y�dy�

�uk���x� � �n��R��k

���x� y�uk�y�dy�

��k ���x� � �n��

R��k

���x� y��k �y�dy� k � �� �� � � � � r�

�����

For �xed � and small enough we have

� dist ���k � �� �k � spt �k�� k � �� �� � � � � r�������

This means that for � and satisfying ��� � and ������ the functions �uk�� and��

k �� vanish near ��k � �� and moreover� they are equal to zero in � n �k�k � �� �� � � � � r� Then� for any �xed � satisfying ��� �� we get���������������������������

k�uk�� � �n��ukkL���� � k�uk�� � �n��ukkL���k� �

� k�uk�� � �n��ukkL����k����� ��

k��k �� � �n���

kkL���� � k��k �� � �n���

kkL���k� �

� k��k �� � �n���

kkL����k����� ��

������

Variational Methods ���

By coordinate transformation we have

�����������������������������������������������

�uk���x� � �n��Z��k

���x� y�uk�y�dy �

Z�k

���x� x�k �z � x�k�

�uk�z�dz �

Z�

���x� x�k �z � x�k�

�uk�z�dz�

��k ���x� �

Z�k

���x� x�k �z � x�k�

��k�z�dz �

Z�

���x� x�k �z � x�k�

��k�z�dz�

������

We put���������u���x� � �u����x� �

rPk��

�uk���x�� x � Rn �

����x� � ������x� �

rPk��

��k ���x�� x � Rn �

Obviously u�� � C�� �Rn �Rn� for all � and satisfying conditions ��� �� �������

Let us �x an arbitrary number � �� We haveZ�

j��u���jdx � supn Z�

��u��� � � dx � � � C�� ��� M n�n

sym �� j� j � � in �o�������

So there exists � � C�� ��� M n�n

sym � and j� j � � in � which depends of course on � �� �� such that

���������

Z�

j��u���jdx �Z�

��u��� � � dx� � � �Z�

u�� � div � dx� �

�rP

k��Ik � ��

������

�����������I� � �

Z�

�u��� � div � dx�

Ik � �Z�

�uk�� � div � dx� k � �� �� � � � � r�

��� FUCHS et al�

We have

I� � �Z�

�u����x� � div ��x�dx � �Z�

u� � �div ���dx �

� �Z�

u� � div ��dx � �Z�

��u � div ��dx�

where

���x� �Z�

���x� y���y�dy� �div ����x� �Z�

���x� y�div ��y�dy� x � Rn �

So we get

I� � �Z�

u�y� � div �������y�dy �Z�

�� � ��dy������

Now let k � f�� �� � � � � rg� Then

Ik � � �

Z�

dxZ�

���x� x�k �z � x�k�

��k�z�u�z�dz � div ��x�dx

� � �

Z�

�k�z�u�z�dzZ�

���x� x�k �z � x�k�

�div ��x�dx

� �

Z�

�k�z�u�z�dzZ�

��ij�x�

�xj���x� x�k �

z � x�k�

��dx�

On the other hand

�xi���x� ��j

��x�kz�x�

k�

� � �

��i���� � x�j

��x�kz�x�

k�

and therefore

Z�

��ij�x�

�xj���x� ��j

��x�kz�x�

k�

�dx �

� �Z�

������i

�� � x�j��x�

kz�x�

k�

�ij�x�dx��

� �div �����j��x�

kz�x�

k�

Variational Methods ���

So we have

Ik � ��

Z�

�k�z�u�z� � div ���x�k �z � x�k�

�dz�

If we introduce the functions

����k� �z� � ���x�k �

z � x�k�

�� z � Rn � k � �� �� � � � � r�

then

����ij�k�

�zj�z� �

���ij�����

��j

�����x�

kz�x�

k�

��

Thus we get

�����������������������

Ik � �Z�

�k�z�u�z� � div ���k �z�dz �

� �Z�

u�z� � div ��k����k� ��z�dz �

Z�

�u�r�k��z� � ���k �z�dz �

� �Z�

u�z� � div ��k����k� ��z�dz �

Z�

�k�z� � ���x�k �

z � x�k�

�dz�

�����

From ������� ������ ����� we get the estimate

�������������

Z�

j��u���jdx � �Z�

u�z� � div����� �

rXk��

�k����k�

��z�dz

�Z�

����z� � ���z� �

rXk��

�k�z� � ���x�k �

z � x�k�

��dz � ��

���� �

We note that

j��j � � in Rn

and therefore

j����k� j � � in Rn � k � �� �� � � � � r�

��� FUCHS et al�

Setting

� � ���� �rX

k��

�k���k �

we get

j�j � � in Rn �������

For �xed � and small enough we have

� � � dist��k� ��k�� k � �� �� � � � � r�������

It is clear that ����k��k � � in � n �k� If z � �� � ��k� then x�k �z�x�

k

� ��

k �

and� by condition ������� ����k� vanishes near ��k � ��� So we can state that

� � C�� ��� M n�n

sym ��

From this and also from ������ ���� �� ������ it follows that�������������

Z�

j��u���jdx �Z�

j��u�j� ��

Z�

h���z� � ���z� �

rXk��

�k�z� � ���x�k �

z � x�k�

�idz�

������

We wish to estimate the last term on the right hand side of ������� We haveZ�

h���z� � ���z� �

rXk��

�k�z� � ���x�k �

z � x�k�

�idz �

�Z�

�� � ��dz �rX

k��

Z�k

�k�z� � ���x�k �

z � x�k�

�dz �

�Z�

� � �����dz �rX

k��

Z�

��x� �Z�k

���x� x�k �z � x�k�

��k�z�dz dx �

�Z�

� � �����dz �nX

k��

�nZ�

��x� �Z��k

���x� y��k �y�dy dx �

�Z�

� � �����dx� �rX

k��

Z�

��x� � ��k ���x�dx� �n

Z�

� �rX

k��

�kdx �

�Z�

j�� � �����jdx� ��� �n�Z�

j��jdx�

��rP

k��

Z�

j��k �� � �n���

k jdx� �nrX

k��

Z�

j�k � �kjdx�

Variational Methods ���

So the �nal estimate takes the form

�����������������������

Z�

j��u���jdx �Z�

j��u�j� � �Z�

j�� � �����jdx�

���� �n�Z�

j��jdx�rX

k��

�Z�

j��k �� � �n���

k jdx�

��nrP

k��

Z�

j�k � �kjdx�

������

It remains to estimate the di�erence�������������������������

Z�

ju�� � ujdx �Z�

j�u��� � u� �rX

k��

�uk�� � ukjdx �

�Z�

j�u��� � u�jdx�rX

k��

�Z�

j�uk�� � �n��ukjdx�

��n��Z�

juk � ukjdx� ��� �n���Z�

jukjdx��

������

Now we proceed in the following way� Let � � �m� There exists �m ���� �� such

that condition ��� � holds and

��� �nm�Z�

j��jdx� �nm

rXk��

Z�

j�mk � �kjdx�

���� �n��m �rP

k��

Z�

jukjdx� �n��m

rXk��

Z�

jumk � ukjdx � �

m�

Next� for � � �m� we determine m according to conditions ������� ������ andsuch thatZ

j�� � �����m jdx� �mrX

k��

Z�

j��mk ��m � �n��m �

mk jdx�

�Z�

j�u���m � u�jdx�rX

k��

Z�

j�umk ��m � �n��m umk jdx � �

m�

Setting um � um��m we get from the last two relations the inequalities

Z�

jum � ujdx � �

m�

Z�

j��um�jdx �Z�

j��u�j� �

m�

��� FUCHS et al�

It follows that

limm��

Z�

jum � ujdx � �

and

lim supm��

Z�

j��um�jdx �Z�

j��u�j�

But since

lim infm��

Z�

j��um�jdx �Z�

j��u�j�

we have established all statements of the lemma� Lemma A���� is proved��

Proof of Theorem A����� The continuity of the imbedding of BD���Rn�into Lp���Rn� for � � p � n

n�� follows from the continuity of the imbeddingof D������ into Lp���Rn� for � � p � n

n�� � Indeed� for any u � BD���Rn��we choose a sequence fumg�m�� as in Lemma A����� For some constant C� �C���� p� n� we have

kumkLp��� � C��kumkL���� � k��um�kL������Passing to the limit we get

kukLp��� � C�kukBD��Rn��

Let us now prove the compactness of the imbedding ofBD���Rn� into L����Rn��Let fusg�s�� be an arbitrary bounded sequence in BD���Rn�� For each us we�nd a sequence fus�mg�m�� as in Lemma A����� For any s � N we can determinea number ms such that

kus�mskD������ � kuskBD��Rn� � ��������

kus�ms � uskL���Rn� � �

s�������

The sequence fus�msg�s�� is precompact in L����Rn� which follows from ������and the compactness of the imbedding of D������ into L����Rn�� On theother hand� ������ implies precompactness of the sequence us in L����Rn��For proving compactness of the imbedding of BD���Rn� into Lr���Rn� weuse the following facts�

Variational Methods ��

i� Bounded sets in BD���Rn� are bounded in Ln

n�� ���Rn��

ii� Sets being bounded in Ln

n�� ���Rn� and precompact in L����Rn� are alsoprecompact in Lr���Rn� for any � � r � n

n�� �

So ����� is proved and it remains to show ������ To do this let us consider theset

V ���� � fu � BD���Rn� �Z�

u � v dx � � �v � V�g�

We �rst prove that there exists a constant C���� n� such that

Z�

jvjdx � C�

Z�

j��v�j �v � V ����������

Suppose that ����� is false� Then we can �nd a sequence vm � V ���� suchthat Z

jvmjdx � mZ�

j��vm�j�

Let vm � vm�R�jvmjdx� Then we have

vm � V �����Z�

j vmjdx � ���

m�Z�

j�� vm�j�

So the sequence vm is bounded in BD���Rn� and we can select a subsequenceconverging to v� � V ���� strongly in L����Rn�� Clearly

Z�

jv�jdx � ��Z�

j��v��j � ��

But from the last relation we see that v� � V� and thus v� � �� This is acontradiction and ����� is proved�

Next let � � r � nn�� � Then we have

kvkLr��Rn� � C���� r� n�h Z�

jvjdx�Z�

j��v�ji�v � V �����

�� FUCHS et al�

and in view of ����� we get that

kvkLr��Rn� � C��C� � ��Z�

j��v�j �v � V ����������

We may introduce some orthobasis in V� in the following way

�����������������

Z�

ek � ej dx � �� k � j� k� j � �� �� � � � � sn�Z�

jekj�dx � �� k � �� �� � � � � sn�

ek � V�� k � �� �� � � � � sn�

For any v � BD���Rn� we have the decomposition

v � v � v�� v � V ����� v� � V��

where

v� �snXi��

ei

Z�

v � ei dx�

From ����� we get

infu��V�

kv � u�kLr��Rn� � kv � v�kLr��Rn� �

� C��C� � ��Z�

j��v � v��j � C��C� � ��Z�

j��v�j�

and Theorem A���� is proved��

Variational Methods ��

A�� Jensen�s Inequality

has been an essential tool in the proof of Theorem ������

LEMMA A���� Let the function ! � R R be dened as

!�t� �

���t�

�dif jtj � d

jtj � d�

if jtj d������

Then� for any u � BD���Rn�� we have the inequality

!� Z��

j��u�j��Z��

!�j��u�j�������

We would like to remark that a version of Lemma A���� can be found in thework �DT��

Proof� According to the de�nition of how to apply a convex function to ameasure we have

Z�

!�j��u�j� � supn�Z�

�u � div � � !��j� j��dx �

� � C�� ��� M n�n

sym �� j� j � �o�

�����

Since

!��t� �

���d�t� if jtj � �

�� if jtj �

we arrive at the representation

Z�

!�j��u�j� � supn�Z�

�u � div � �d

�j� j��dx �

� � C�� ��� M n�n

sym �� j� j � � in �o�

�����

��� FUCHS et al�

Let the numbers �� m � N satisfy the condition

��

m�����

We put as usual

u��x� �Z�

���x� y�u�y�dy

where

���x� ��

n��jxj �� ��t� � cn

��� e�

t��� if jtj � �

� if jtj � ��

and cn is chosen so thatZRn

���x�dx � ��

We have

u� u in L����Rn������

Let us introduce the family of sets

�m � fx � � � dist���� x� �

mg

and set

� �

����d��u�� if j��u��j � d

��u��j��u��j if j��u��j d�

��� �

It is clear that � � L����Rn� and j� j � � a�e� in �� We also introduce thetruncation of � by letting

�m �

������ if x � �m

�� x � � n �m�

Variational Methods ���

Direct calculations show that

�������������

Z�m

!�j��u��j�dx �Z�m

���u�� � �m � !��j�mj��dx

�Z�m

���u�� � �m � d

�j�mj��dx�

�����

On the other hand� for each m� a sequence f�mk g�k�� � C�� ��m� M

n�nsym � exists

such that

�����j�mk j � � in �m�

�mk �m in L���m� Mn�nsym ��

�����

Therefore

Z�m

���u�� � �mk � d

�j�mk j��dx k���

Z�m

!�j��u��j�dx�

So for any � � we can �nd some � � C�� ��m� M

n�nsym � such that j�j � � in �

and

Z�m

!�j��u��j�dx �Z�m

���u�� � � � d

�j�j��dx� ��������

Next we have

�����������������������

Z�m

��u�� � � dx � �Z�m

u� � div� dx

� �Z�

u�y� �Z�m

���x� y�div ��x�dx dy

� �Z�

u�y� �Z�

���x� y�div ��x�dx dy � �Z�

u�y� � div ���y�dy�

������

where �as usual�

���y� �Z�

���y � x���x�dx �Z�m

���y � x���x�dx�

��� FUCHS et al�

and� by condition �����

�� � C�� ��� M n�n

sym �� j��j � � in ��������

We observe thatZ�

j��j�dx �Z�

j�j�dx �Z�m

j�j�dx�

From this and also from ������� ������ we derive

�������������

Z�

!���u���dx � �Z�

�u � div �� � d

�j��j��dx� �

�Z�

!�j��u�j� � ��������

Next� since the function ! is convex and di�erentiable� we have

!�j��u���x�j�� !� Z��m

j��u��jdx��

� !�� Z��m

j��u��jdx��j��u���x�j �

Z��m

j��u��jdx��

Integrating this inequality and using ������ gives

j�mj!� Z��m

j��u��jdx��Z�

!�j��u�j� � ��

Passing to the limit as � and taking into account ����� we obtain

lim inf���

Z��m

j��u��jdx �Z��m

j��u�j

and

j�mj!� Z��m

j��u�j��Z�

!�j��u�j� � ��

Variational Methods ���

Using � ��Sm��

�m� �m � �m�� we get

limm��

Z��m

j��u�j �Z��

j��u�j�

and therefore

j�j!� Z��

j��u�j��Z�

!�j��u�j� � ��

By the arbitrariness of � we have completed the proof of the lemma��

Chapter �

Quasi�static �uids of

generalized Newtonian type

��� Preliminaries

Let us brie�y review the physical background of the problems under consi�deration� we discuss the �ow of incompressible generalized Newtonian �uidswhose behaviour is characterized in terms of di�erential inequalities� To beprecise� let us assume that the �uid occupies a region � � R

n �i�e� a boundedLipschitz domain�� n � �� we denote by v � v�x� t�� x � �� the velocity �eldand since the �uid is incompressible� we have

div v � �������

Note that by the mass balance law ����� just expresses the fact that the densityis a constant function� For a given system f � � R

n of volume forces �whichare assumed to be independent of time� the �uid has to obey the equation ofmotion

div � � f � 'v������

Here � is the �apriori unknown� stress tensor ��ij� and 'v denotes the materialderivative� i�e� the quantity �rv�v � �tv with the vector function �rv�v ���iv

jvi���j�n� In addition we have to impose

various boundary and initial condition for v and �������

���

Variational Methods ���

Our main assumption is that the velocity should not depend on t �the time�dependent situation is discussed in the papers �L���� of Ladyzhenskaya and

in the recent book of M�alek� Nec�as� Rokyta� R�u�zi�cka �MNRR��� In this case

����� has to be replaced by

������ div � � f � �rv�v�

where now v � v�x�� � � ��x� and ����� reduces to boundary conditions forv and � � Following standard convention �see �DL�� we adress this situation asthe stationary or quasi�static case� In the case of slow� steady state motionequation ������ may be replaced by

������ div � � f � ��

Of course the assumption �rv�v � � seems to be arti�cial but� from the ma�thematical point of view� as we will show later on� sometimes the quasi�staticcase can be reduced to this situation� In order to characterize the speci�c �uidunder consideration we need a so�called constitutive law which connects thestress tensor � and the strain velocity ��v� � �

���iv

j � �jvi�� For various clas�

ses of generalized Newtonian �uids and in particular viscoplastic �uids such aconstitutive relation can be formulated with the help of a so�called dissipativepotential W � M R which is a given convex function de�ned on the spaceof all matrices of order n� a physical discussion is given for example in �AM���BAH�� �DL�� �IS�� �L����� �MM��� �MNRR�� �Pr� and �Ser��

We formulate this stress�strain relation as

�D � �W ���v��������

�D � � � �ntr �� �stress deviator�� Here �W is the subdi�erential of the

convex function W � for A�B � M we have A � �W �B� if and only if W �T � �W �B��A � �T�B� for all T � M � If the dissipative potential is a di�erentiablefunction� then ����� is just another formulation of the equation �D � �W

�E���v���

A typical example for a nonsmoothW occurs in the case of a classical Bingham�uid� with positive constants � and k� we have �see �DL�� �MM���

W �E� � �jEj� �p�k�jEj� E � M �

��� FUCHS et al�

and from ����� we get the equation

��v� �

�����������

p�k�j�Dj ��

D if j�Dj � p�k�

� if j�Dj � p�k�

which means that for small stresses the �uid behaves like a rigid body �i�e���v� � ��� The problem under investigation can be formulated as follows�

�����to �nd a pair �v� �� of functions such that ������ ������ �or �������� �����

and ����� are satis�ed�

For simplicity let us assume that we have �xed Dirichlet boundary data forthe unknown velocity �eld� i�e� v has to be found in the class K � fw � � Rn jdiv w � �� w � v� on ��g with given function v�� Let us further assume

that �v� �� denotes a pair of smooth solutions under the hypothesis �������Then it is easy to show that the velocity �eld v minimizes the variationalintegral

J�v� �Z�

�W ���v��� f � v� dx

among functions in K � and it is therefore reasonable to study the variationalproblem

J min in K����

separately� As far as the existence of �weak� solutions is concerned we mayargue along standard lines� the dissipative function W is convex with ap�propriate growth rates so that theorems on lower semicontinuous variationalintegrals together with Korn type inequalities show solvability of ���� provi�ded the class K is imbedded in a suitable Sobolev space� In order to come backto the original problem we then have to analyze the regularity properties ofsolutions to ����� for example we ask if minimizers are of class C�� This willbe done in the next section� Unfortunately� apart from the two�dimensionalcase n � �� we only obtain partial C��regularity which means that minimizersare smooth outside a �closed� singular set with small measure� and there is nohope to improve these results� in the Bingham case for example the dissipativepotential is irregular at � � M which means that C��regularity of a minimizer

Variational Methods ��

v can only be expected on the set ���v� � ��� But even if the potential Wis smooth but non�quadratic� the experience in regularity theory �see �Gi��suggests that singularities may occur in dimensions n � ��

The next results concern local boundedness of the strain velocity� for variousclasses of generalized Newtonian �uids including the Bingham model we provein section � that the tensor ��v� is in the space L�loc��� M � which implies H&oldercontinuity of v for any exponent � � � � �� Moreover� we will show that theclassical derivative ru�x� exists for almost all x � ��

Up to now all our discussions are limited to the variational setting which meansthat we start from the assumption �rv�v � � which leads us to the variationalproblem ����� It is worth remarking that the results described above are validin any dimension n� moreover� as a byproduct of our investigation we obtaintheorems for certain nonlinear Stokes systems� In fact� if the dissipative func�tion W is of class C�� then local J�minimizers v with respect to the constraintdiv v � � are solutions of

Z�

�W

�E���v�� � ���� dx �

Z�

f � �dx� � � C�� ���Rn�� div � � ������

which for W �E� � ��jEj�� E � M � reduces to the classical Stokes system

studied for example in �L��� if the dissipative potential is quadratic� then

Z�

�W

�E���v�� � ���� dx �

Z�

rv � r�dx

for solenoidal test vectors � and ���� turns into the set of equations

�����div v � �

���"v � f �rp

with a suitable pressure function p�

Let us now discuss the exact equation of motion ������� By the same reasoningas described above we can show� if �v� �� are smooth solutions of ������ ������

�� FUCHS et al�

and ����� and if for simplicity we just impose Dirichlet boundary conditionson v� then v is a solution of the problem

�������������������������

div v � ��

Z�

�W

�E���v�� � ���� � �rv�v � �dx �

Z�

f � �dx

for all � � C�� ���Rn�� div � � ��

v � v� on ��

��� �

provided W has continuous �rst derivatives� In the general case of nonsmoothW ��� � has to be replaced by a variational inequality which for the Binghammodel �w�l�o�g� � � ����

p�k� � �� reads

���������������������������

to �nd v � K such that

Z�

��v� � ���w�� ��v�� � j��w�j � j��v�j� �rv�v � �w � v� dx �Z�

f � �w � v� dx

holds for all w � K �

�����

Equation ��� � is just an extension of the time independent Navier�Stokessystem �see �L�� or �Ga���

�����div v � � � v � v� on ���

���"v � �rv�v � f �rp

which is easily derived from ��� � by letting W �E� � ��jEj�� E � M � In

contrast to the variational case the existence of weak solutions for ��� � or����� in suitable energy spaces is no longer obvious� in fact� we are confrontedwith the same di�culties as for the stationary nonlinear Navier�Stokes system�Following the lines of �L�� and �DL� we get at least the existence of generalizedsolutions for homogeneous boundary data in dimensions n � � and � usingelementary arguments� But since we concentrate on regularity properties� wewill just assume that we already have some local weak solution of ��� � and

Variational Methods ��

������ Fortunately most of the regularity results obtained in the variationalsetting extend to solutions of ��� � and ������ let us de�ne the arti�cial volumeload f � f � �rv�v� where v solves ��� � or ����� in the weak sense� Then v islocal minimizer of

w Z�

�W ���w��� f � w� dx

subject to the constraint div w � �� For proving regularity in the variationalsetting it is su�cient to know that the volume forces f belong to some Morreyspace which means that now we have to analyze the nonlinearity �rv�v� Indimensions n � � and � we will show some growth properties of

RBr�x��

jrvj�dx�Br�x�� � �� from which we derive that f is in the �right� Morrey class� Fordetails we refer to section ��

Next we give some important examples of dissipative potentialsW whose phy�sical motivation can be found in �AM�� �BAH�� �DL�� �IS�� �MM��� �Ka�� �Ki�and �Pr��

a� �uids of Bingham type� we let

W �E� ��

mjEjm � �jEj� E � M �

with constants m �� � � �� As subcases this model includes

i� Newton �uids� m � �� � � �In this case ���� is reduced to the Stokes system� whereas ��� � isthe stationary Navier�Stokes system�

ii� classical Bingham �uids� m � �� � �

iii� Norton �uids� m �� � � � �see �N��

b� power law models� for nonnegative real numbers ��� �� such that�� � �� � we let

W �E� � ��jEj� � ��

������� � jEj��p� � �� p � �

orjEjp � p �

��� �

��� FUCHS et al�

c� Powell�Eyring model� according to �PE� we de�ne

W �E� � ��jEj� � ��

Z jEj

�ar sinh t dt� E � M �

with constants ��� �� ��

d� Prandtl�Eyring model� passing to the limit �� � in the previous casewe get the dissipative potential

W �E� � ��

jEjZ�

ar sinh t dt� E � M �

which can be found in the work �E��

As the reader might guess from the de�nition� the last model requires separatemethods� for example� we have to de�ne appropriate function spaces� For thisreason Prandtl�Eyring �uids are postponed to chapter �� We wish to remarkthat a condensed version of chapter � can be found in the paper �Fu� whereasa more detailed version was presented in �Fu���

We �nish this section by recalling some auxiliary results we shall use later�

Let G � Rn denote a smooth bounded region and suppose that p ������ is a

given real number� We use the symbol c to denote various positive constantsdepending on n� p and the domain G�

The �rst lemma is an extension of the classical Korn�s inequality �see �Ko �����for a proof and further references see also �Fi�� �Fri�� �Str�� �Z���

LEMMA �� �� Suppose u � Lp�G�Rn� and ��u� � ���ru�ruT � � Lp�G� M ��

Then u belongs to the Sobolev space W �p �G�Rn� and we have the estimate

kukW �p� cfkukLp � k��u�kLpg�

For v � �

W �p�G�Rn� the inequality

kvkW �p� ck��v�kLp

holds�

Variational Methods ���

We refer the reader to �MM�� �see also �IS�� �Fu����

Sometimes we make use of the following interpolation inequality

LEMMA �� �� For any function u � W �p �G�Rn� with p � � we have kukLp �

cfkukL� � k��u�kLpg�

This result can easily be obtained by contradiction using Lemma ������

Let V� denote the space of rigid motions� i�e� u � V� has the form u�x� �a�Bx with a � Rn and B � M skew�symmetric�

LEMMA �� �� i� A function u � W �p �G�Rn� belongs to the space V� if and

only if ��u� � ��

ii� For all u � W �p �G�Rn� we have inf

r�V�ku� rkLp � ck��u�kLp�

iii� The inmum in ii� is achieved� i�e� for each u � W �p �G�Rn� we nd r � V�

such that ku� rkLp � ck��u�kLp�

In order to construct suitable comparison functions we make essential use of

LEMMA �� �� Suppose f � Lp�G� with �RGf dx � �� Then there exists a

function u � �W �

p�G�Rn� satisfying

div u � f and

� krukLp � ckfkLpkrukL� � ckfkL��

Proof� cf� �LS�� �P�� �Ga��� III� Theorem ���� �Fu��� proof of Lemma ��

Next we collect various Campanato�type estimates for the linear Stokes sy�stem which are rather well�known but di�cult to trace in the literature�

�� FUCHS et al�

LEMMA �� �� Let A denote a symmetric strictly positive bilinear form onthe space of all symmetric �n � n� matrices� Then there is a constant c� �c��n�A� independent of G such that for v � W �

� �G�Rn� satisfying

�����������div v � �

ZG

A���v�� ����� dx � �� � � �W �

��G�Rn�� div � � �

we have v � C��G�Rn� and further �here BR � BR�x�� � G��

i�Z

BR��

j��v�j� dx � c�R��ZBR

jv � u�j� dx� u� � V��

ii�Z

BR��

jrvj� dx � c�R��ZBR

jv � �j� dx� � � Rn �

iii�ZBr

jrvj� dx � c��r

R�nZBR

jrvj� dx� Br � BR�

iv�ZBr

jv � �v�rj� dx � c��r

R�n�

ZBR

jv � �v�Rj� dx�

v�ZBr

j��v�� ���v��rj� dx � c��r

R�n�

ZBR

j��v�� ���v��Rj� dx and

vi�ZBr

jrv � �rv�rj� dx � c��r

R�n�

ZBR

jrv � �rv�Rj� dx�

Proof� i� �cf� �GM�� Thm����� for a similar argument�� Let � denote a cut�o�function satisfying � � � on BR�� spt � � BR� jr�j � ��R� � � � � �� andde�ne �� �� ���v � u��� Then

div �� � �� r� � �v � u��

Variational Methods ��

and according to Lemma ����� we can �nd w � �W �

��BR�Rn� satisfying

div w � �� r� � �v � u���ZBR

jrwj� dx � c�ZBR

jr� � �v � u��j� dx�

� �� �� � w is in�W �

��BR�Rn� with div � � �� hence

ZBR

A���v�� ����� dx � ��

Observing ���� � ����� � ��w� � ����v� � �fr� � �v � u�� � �r� � �v �u���Tg � ��w� we arrive at

ZBR

��A���v�� ��v�� dx � c�n ZBR

�j��v�jjr�jjv� u�j dx�ZBR

j��v�jj��w�j dxo�

From Young�s inequality we deduce for small positive �

ZBR��

j��v�j� dx � �ZBR

j��v�j� dx� c����R��ZBR

jv � u�j� dx�

With obvious modi�cations �replacing R�� by some radius s and R by t takenfrom �R��� R�� the result follows from �GM�� Lemma ���

To see ii� we combine i� and Lemma ����� �the dependence on the radius ofthe constants occuring in Korn�s inequality is easily checked by scaling��

In the next step one shows v � W k��loc�G�Rn� for all k � N � thus v � C��G�Rn�

and any partial derivative of arbitrary order is a solution of the constant�coe�cient problem� It is then easy to proceed along the lines of �Gi� using i��ii� from above for the derivatives of v by the way proving iii��vi��

For the readers convenience we show v � W ���loc�G�Rn�� The main ideas are

taken from �GM�� proof of Theorem ���� Choose a ball BR � G and de�ne �as in i�� Let

�� �� ��h��� �hv�

�� FUCHS et al�

where �hw�x� ���h�w�x�he���w�x�� for some �xed direction e� � � � �� � � � � n�

We then de�ne � �� �� � w with w � �W �

��BR�Rn� such that div w � div ��

and

krwkL��BR� � c�kdiv��kL��BR��

�Of course we assume jhj to be su�ciently small so that spt �� � BR��

We obtainZBR

A���v�� ����h����hv��� dx �

ZBR

A���v�� ��w��dx�

The �rst integral equalsZBR

A����hv�� �����hv�� dx � c�

ZBR

��j�h��v�j�dx� c�ZBR

j���hv�j�jr�jj�hvj dx

and with Young�s inequality we deduceZBR

��j���hv�j�dx � c�fZBR

jr�j�j�hvj�dx�ZBR

j��v�jj��w�j dxg�

We next observeZBR

j��w�j�dx �ZBR

jrwj�dx � c�ZBR

j��h�div����hv��j�dx

� c�ZBR

j��h���r� � �hv�j�dx

� c�ZBR

j��h��r��j�j�hvj�dx� c�ZBR

��jr�j�j��h��hv�j�dx�

here we have used the fact that v is divergence�free� From H&older�s inequalitywe have Z

BR

j��v�jj��w�j dx � c���R��ZBR

j��v�j�dx� �R�ZBR

j��w�j�dx�

Since ZBR�jhj

j�hwj�dx �ZBR

jrwj�dx

Variational Methods ��

we end up withZBR��

j���hv�j�dx � �ZBR

j�hrvj�dx � c���R��ZBR

jrvj�dx�

Finally we apply Korn�s inequality to the function �hv yieldingZBR��

jr��hv�j�dx � c�fZ

BR��

R��j�hvj�dx�Z

BR��

j���hv�j�dxg�

Combining the last two inequalities we haveZBR��

jr��hv�j�dx � �ZBR

jr��hv�j� dx� c���R��ZBR

jrvj� dx�

and the claim follows from Lemma �� in �GM��

Let us indicate how to obtain v�� First of all� for any weak solution u to theabove system� we have inequality vi�� i�e�Z

Br

jru� �ru�rj�dx � c�� rR

�n�ZBR

jru� �ru�Rj�dx�

This is a consequence of iv�� we just have to replace the function by somearbitrary derivative� iv� �and in the same way iii�� in turn can be deducedfrom i�� ii� and the W k

��loc�regularity along the lines of �GM�� Proposition ����

Assume now r � R�� and let �x� � �rv�r�x�� ThenZBr

j��v�� ���v��rj�dx �ZBr

j��v � �j�dx �

ZBr

jr�v � �j�dx �ZBr

jrv � �rv�rj�dx ��vi�

c�Z

BR��

jrv � �rv�R�j�dx� rR

�n� �

c��rR

�n�Z

BR��

jrv � Aj�dx

�� FUCHS et al�

for any n�n matrix A� Let w�x� � v�x�� ���v��Rx and choose a rigid motion��x� such that

ZBR

jw � �j�dx � c�R�ZBR

j��w�j�dx�

Let us also set u � w � �� A � ���v��R �r�� This givesZ

BR��

jrv � Aj�dx �Z

BR��

jruj�dx �

c�R��ZBR

juj�dx�

the last inequality follows from ii� applied to u� Putting together the estimateswe �nd thatZ

BR��

jrv � Aj�dx � c�ZBR

j��w�j�dx�

here ZBr

j��v�� ���v��rj�dx � c�� rR

�n�ZBR

j��v�� ���v��Rj�dx

and v� is established��

Variational Methods �

��� Partial C� regularity in the variational

setting

In �Se������ partial regularity results for the variational problem ���� in caseof a Bingham �uid� i�e� W �E� � �jEj� � p

�k�jEj� were obtained� Later on�see �FGR�� this result was extended to general �uids of Bingham type� Wehave the following

MAIN THEOREM ����� Suppose we are given a real number p � �� afunction v� � W �

p ���Rn� such that div v� � � and consider a volume load f� in

Lp���Rn� which belongs to some Morrey space Lp��n�p�p�loc ���Rn�� � � � � ��

p� � p�p� �� Then the variational problem

�������I�u� �

Z�

�j��u�jp � j��u�j � f� � u� dx min in

K � fv � W �p ���R

n� � div v � �� v � v� on ��g

admits a unique solution u� Dene the sets

�� � fx � � � x is a Lebesgue point for ��u� and

limr��

Z�

Br�x�

j��u�� ���u��x�rjpdz � �g�

�� � fx � �� � ��u��x� � �g�

Then we have the following statements�

a� �� is open and ��u� � C������� M � for any � � ��

b� ru � C������� M � for any � � ��

Moreover� ��u� � � a�e� on �� ���

The proof of the theorem �taken from the paper �FGR�� is organized in severalsteps� for technical simplicity we may assume that the volume forces f� vanish�

We �rst consider a more regular functional of the form

J�u� �Z�

j��u�jp � f���u�� dx

� FUCHS et al�

with f smooth� convex but growing of order less than p� Let u denote a J�minimizer in the class K � existence and uniqueness of such u �and indeed forminimizers of the original problem� follow by standard methods� see e�g� �Gi��We show that ��u� is H&older�continuous near a point x� � � provided x� is aLebesgue�point for ��u�� ��u��x�� � � and

limr��

Z�

Br�x��

j��u�� ���u��xo�rjp dz � ��

The main ingredient here is a blow�up Lemma which for general p � � is mo�re complicated than in the quadratic case� Next we replace f by a sequence

f��Q� ��q�� � jQj� with corresponding J��minimizers u�� It is not hard to

show that u� u as � � � strongly in W �p where u is the solution from the

Main Theorem� Moreover� by the �rst step� we have uniform partial regularityfor ��u�� which gives part a� of the Main Theorem by letting � tend to ��

To be precise� let

J�u� �Z�

j��u�jp � f���u�� dx

with f � M ����� of class C� and convex such that the growth condition

jD�f�Q�j � a� b�� � jQj�� q��������

holds for constants a� b and q such that a� b �� � � q � p� A typical exampleis f�Q� � ��� � jQj��q�� � � �� We have the following result analogous to theMain Theorem �we use the notation of that theorem��

THEOREM ����� The functional J admits a unique minimizer u in theclass K � This minimizer satises�

a� �� is open� and ��u� � C������� for all � such that � � � � ��

b� ru � C������� for all � such that � � � � ��

The key element in the proof of Theorem ����� is the growth estimate of Lemma����� �cf��EG�� section �� or �FR��� The proof of Theorem ����� given Lemma����� is standard� the arguments are a simpli�cation of those used in the proofof the Main Theorem given after Lemma �����

Variational Methods �

LEMMA ����� �Blowup lemma� Let A be a non zero symmetric matrixwith norm ��� and let u be a local minimizer for J in K � Then there exists aconstant c� depending only on A and C� bounds on f in the � neighborhoodof A such that for every t ���� �� there exists � depending only on t� f and Asuch that the conditions

j���u��x��R � Aj � �������

E�u�BR�x��� ��

Z�BR�x��

j��u�� ���u��x��Rjp dx �Z�BR�x��

j��u�� ���u��x��Rj� dx � �������

for some ball BR�x�� � � together imply the estimate

E�u�BtR�x��� � c�t�E�u�BR�x���������

Proof� If the proposition were false we could �nd� for any c� � andt ���� ��� a sequence of balls fBRk�xk�g � � and a sequence fvkg of localminimizers for J in K such that the estimates

j���vk��xk�Rk � Aj � ������

E�vk� BRk�xk�� �� ��k � � as k �����

and

E�vk� BtRk�xk�� c�t���k��� �

were all valid�

Set Ak �� ���vk��xk�Rk � and de�ne gk�z� � vk�xk � Rkz� � RkAkz� for z � B��It is immediate that gk � W �

p �B��Rn�� so by Lemma ����� �iii� there exists

pk � V� such thatZB�

jgk�z�� pk�z�j� dz � c�

ZB�

j��gk�j� dz�����

�� FUCHS et al�

for c� depending only on n and p� De�ne now

uk�z� ���

Rk�k�gk�z�� pk�z���

We see by ����� �using also �����

�����������������

ZB�

jukj� dz � c��R�k��k

ZB�

j��gk�j� dz

�c��

Rnk��k

ZBRk

j��vk��y�� ���vk��xk�Rk j� dy

� c���n�

�����

where �n denotes the volume of the unit ball� Further we haveZ�B�

j��uk�j� dz � ��������

Z�B�

j��uk�jp dz � ���pk �������

We thus have the existence of v � W �� �B��R

n� and A� � M such that� afterpassing to an appropriate subsequence�

uk � v in W �� �B��R

n�������

uk v in L��B��Rn�������

����pk ��uk� � � in Lp�B�� M �������

and

Ak A�������

here ������ follows from ������ ������ and Korn�s inequality� ������ follows from������ and ����� follows from ����� To see ������ we note from ������ thereexists h such that ����p��uk� � h in Lp� for any � � C�

� �B� M � we thus have

���ZB�

h � � dz��� � lim

k��

���ZB�

����pk ��uk� �

� dz��� � lim sup

k������pk k��uk�kL��B��k�kL��B�� � �

Variational Methods ��

by ���� and �������

Assume now the conclusion of Lemma ������ viz� that the weak convergencein ������ and ������ can be improved to strong convergence �after passing toappropriate subsequences�� Assume further Lemma ������ the blow�up equa�tion� and set c� �� �c�� where c� is the constant in Lemma ���� correspondingto the bilinear form associated with equation ������� note that c� depends onlyon A and C� bounds on f in the ��neighborhood of A� Lemma ���� �v� thensays

Z�Bt

j��v�� ���v��tj� dz � c��t�Z�B�

j��v�� ���v���j� dz������

On the other hand� rewriting ��� � in terms of the rescaled map uk yields

E�uk� Bt� c�t��

which by Lemma ����� then implies

Z�Bt

j��v�� ���v��tj� dz � c�t������ �

We have from ������ and ������

Z�B�

j��v�j� dz � lim infk��

Z�B�

j��uk�j� dz � �

which� combined with ���� �� yields

Z�Bt

j��v�� ���v��tj� dz � c�t�Z�B�

j��v�� ���v���j� dz�

comparing this to ����� gives the desired contradiction��

We next establish the auxiliary results discussed above� viz� the blow�up equa�tion and the desired strong convergences�

�� FUCHS et al�

LEMMA ����� �Blowup equation� With notation as above� the limit map v satises the equation

ZB�

pjA�jp�����v� � �p� ��jA�j���A� � ��v��A�� � ���� dz

� �Z

B�

��f

�Q��A�����v�� ����� dz

������

for all � � C���B��R

n� such that div � � ��

Proof� From the Euler�Lagrange equation for vk we have

ZBRk �xk�

pj��vk�jp����vk� � ���� dy � �Z

BRk �xk�

�f

�Q���vk�� � ���� dy

for all � � C�� �BRk�xk��R

n� such that div � � �� Rewriting this in terms ofthe rescaled map uk yields

ZB�

pj�k��uk� � Akjp����k��uk� � Ak� � ���� dz

� �Z

B�

�f

�Q��k��uk� � Ak� � ���� dz

������

for all � � C�� �B��R

n� such that div � � �� for the remainder of the proof� we�x such a �� Noting

RB�

jAkjp��Ak � ���� dz � �� we subtract this quantity from

the left�hand side of ������� and similarly subtract � �RB�

�f�Q

�Ak� � ���� dz

from the right�hand side� multiplying through by ���k then yields � � Ik � IIk�

where Ik �� p ���k

ZB�

�j�k��uk��Akjp����k��uk��Ak��jAkjp��Ak� � ���� dz

and IIk �� ���k

ZB�

��f

�Q��k��uk� � Ak�� �f

�Q�Ak�� � ���� dz�

Variational Methods ��

We rewrite

Ik �� p ���k

ZB�

�Z�

d

ds�jAk � s�k��uk�jp���Ak � s�k��uk��� jAkjp��Ak� ds �

���� dz

� I�k � I�k �

where I�k �� pZB�

�Z�

jAk � s�k��uk�jp�� ds ��uk� � ���� dz�

I�k �� p�p� ��RB�

�R��jAk � s�k��uk�jp���Ak � s�k��uk�� �

��uk���Ak � s�k��uk�� ds � ���� dz�

Similarly we have

IIk � ���k

ZB�

�Z�

d

ds

��f

�Q�Ak � s�k��uk��� �f

�Q�Ak�

�ds � ���� dz

�ZB�

�Z�

��f

�Q��Ak � s�k��uk�����uk�� ����� dsdz�

The desired equation ������ follows directly� once we can establish the followingasymptotic behaviour �possibly after passing to a subsequence��

limk��

I�k � pjA�jp��ZB�

��v� � ���� dz�������

limk��

I�k � p�p� ��jA�jp��ZB�

�A� � ��v��A� � ���� dz� and������

limk��

IIk �ZB�

��f

�Q��A�����v�� ����� dz�������

From ������ and ������ equation ������ is immediate once we establish

limk��

I�k � ��������

�� FUCHS et al�

where

I�k ��ZB�

�� �Z�

jAk � s�k��uk�jp�� ds� jAkjp���� ��uk� � ���� dz�

To see this� we begin by noting� from ������

�k��uk� � in Lp� with pointwise convergence a�e� in B��������

Given � �� we can thus apply Egorov�s theorem to �nd M � M��� � B�

with jM j � � such that �k��uk� � uniformly on B� nM � We then estimate

jI�k j � k����k� supB�nM

��� �Z�

jAk � s�k��uk�jp�� ds� jAkjp����� ZB�nM

j��uk�jdz

�k����k�ZM

��� �Z�

jAk � s�k��uk�jp�� ds� jAkjp�����j��uk�j dz�

The �rst term tends to zero as k� by ������ and the uniform convergenceof �k��uk� to zero on B� nM � To estimate the second term� we note by ����

������ � �kj��uk�jjAkj

�����p��

� c��� � �p��k j��uk�jp���

for c� �� �p��maxf���p� �g�

We thus have

ZM

��� �Z�

jAk � s�k��uk�jp�� ds� jAkjp�����j��uk�j dz

� jAkjp��ZM

h�� �

�kj��uk�jjAkj �p�� � �

ij��uk�j dz

� �p���p���c� � ���ZM

j��uk�jdz � �p��k

ZM

j��uk�jp�� dz�

� �p���p���c� � ���p�� ��p�

���pk ��

Variational Methods ��

the last line following from H&older�s inequality� ������ and ������� By ���� andthe arbitrariness of �� this establishes ������� and hence �������

In a completely analogous manner we derive ������ and ������� we use thestructure condition ����� in showing ������� By the remarks above� this com�pletes the derivation of �������

It remains to show

LEMMA ����� �Strong convergence� The convergence properties statedin ������� ������ can be improved to strong convergence� i�e�

��uk� ��v� in L�loc�B�� M �

����pk ��uk� � in Lp

loc�B�� M ��

Proof� As in �EG� �see also �FR�� we de�ne the scaled energy density

fk�Q� � ���k �jAk � �kQjp � jAkjp � pjAkjp��Ak � �kQ�

����k �f�Ak � �kQ�� f�Ak�� �f�Q

�Ak� � �kQ�� Q � M �

For � � r � � we let

Irk�w�Br� ��ZBr

fk���w��dz

for any w � W �p �Br�R

n� with div w � �� In view of the original problem it isobvious that uk is a minimizer of Irk�� � Br�� i�e�

ZBr

fk���uk��dz �ZBr

fk��� u�� dz�����

whenever u � W �p �B��R

n�� spt �uk � u� �� Br� div u � ��

�� FUCHS et al�

We claim

� � fk�Q� � c��jQj� � �p��k jQjp� �Q � M�����

for some positive constant c� independent of k but depending on the sameparameters as c� from Lemma ������ This follows from

���k �jAk � �kQjp � jAkjp � p�kjAkjp��Ak � Q�

� p

�Z�

njAk � s�kQjp��Q � Q � �p� ��jAk � s�kQjp��

�Q � �Ak � s�kQ���o��� s� ds

� p �p��k

�Z�

n���Ak

�k� sQ

���p��Q � Q� �p� �����Ak

�k� sQ

���p��hQ �

�Ak�k

� sQ�i� o

��� s� ds

� c� �p��k jQj�

�Z�

���Ak

�k� sQ

���p�� ds� c� �

p��k jQj�

����Ak�k

���p�� � jQjp���� c� �jQj� � �p��k jQjp�

���� �

and �using the growth condition ����� imposed on f�

���k �f�Ak � �kQ�� f�Ak�� �f�Q

�Ak� � �kQ�

��Z

��� ��f�Q�

�Ak � s�kQ����jQj���� s� ds

��Z

jQj��a� b�� � jAk � s�kQj��

q���

���� s� ds

� c�jQj��� � �p��k jQjp���Following Evans�Gariepy �EG� we de�ne the measures

�k�Z� ��ZZ

j��uk�j� � �p��k j��uk�jp dx� Z � B�����

Variational Methods �

According to ������ and ������ the sequence �k is uniformly bounded� and thuswe have the existence of a Radon measure � on B� such that� after passing toan appropriate subsequence� �k � ��

We choose � � r � � such that ���Br� � �� Let � � s � r� We de�ne

� ��

��� v on Bs���

# � uk � ��v � uk� on Br���� Bs���

where � � C���Br� is a cut�o� function chosen to satisfy � � � � �� � � � on

Bs� jr�j � c�r�s � and # � �

W �p�Br � Bs�R

n� is a solution of

div # � �div �uk � ��v � uk���

According to Lemma ����� # satis�es �t � �� p�ZBrnBs

j��#�jtdx �Z

BrnBsjr#jtdx � c��r� s�

ZBrnBs

jr��v � uk�jtdx�������

Combining ������ ����� and ������ we deduce���������������������������������������������������������������������������������

Irk�uk�� Irk�v� � Irk�# � uk � ��v � uk��� Irk�v�

�Z

BrnBsfk����# � uk � ��v � uk��

�dx

� c�

ZBrnBs

nj��# � uk � ��v � uk��j� � �p��k

j��# � uk � ��v � uk��jpodx

� c�

ZBrnBs

nj��#�j� � j��uk�j� � jr�j�jv � ukj� � j��v�j�

��p��k �j��#�jp � j��uk�jp � jr�jpjv � ukjp � j��v�jp�odx

� c��

ZBrnBs

fj��uk�j� � �p��k j��uk�jp � j��v�j� � �p��k j��v�jpg dx

�c���r� s�Z

BrnBsfjv � ukj� � �p��k jv � ukjpg dx�

������

� FUCHS et al�

Now we consider the last integral� We already know

ZBrnBs

jv � ukj� � �� as k ��

On the other hand we have by Lemma �����

k����pk �v � uk�kLp�BrnBs� � c���r� s�fk����pk �v � uk�kL��BrnBs�

�k������pk �v � uk��kLp�BrnBs�g�

Both terms on the right hand side are uniformly bounded� Applying Korn�sinequality� Lemma ������ we obtain

k����pk �v � uk�kW �p �BrnBs� �

c���r� s�fk����pk �v � uk�kL��BrnBs� � k����pk ��v � uk�kLp�BrnBs�g�

Since ����pk �v � uk� � in L��Br nBs� we end up with

����pk �v � uk� � strongly in Lp�Br nBs��

We now pass to the limit k � in ������ with the result

lim supk��

�Irk�uk�� Irk�v�� � c��f��Br nBs� �Z

BrnBsj��v�j� dxg�

Taking the limit s� r we arrive at

lim supk��

�Irk�uk�� Irk�v�� � � for almost all r � ��� ���������

We next claim

lim supk��

�Irk�uk�� Irk�v�� �

c�� lim supk��

ZBr�j��uk�� ��v�j� � �p��k j��uk�� ��v�jp� dx

������

Variational Methods �

for some positive constant c��� Inequality ������ follows from

Irk�uk�� Irk�v� �

ZBr

�fk���uk��� fk���v��� dx

� ���k

ZBr

jAk � �k��uk�jp � jAk � �k��v�jp � pjAkjp��Ak �

�k���uk�� ��v�� dx

����k

ZBr

f�Ak � �k��uk��� f�Ak � �k��v��� �f

�Q�Ak� �

�k���uk�� ��v�� dx

� ���k

ZBr

jAk � �k��uk�jp � jAk � �k��v�jp � pjAk � �k��v�jp��

�Ak � �k��v�� � �k���uk�� ��v�� dx

����k

ZBr

�p�jAk � �k��v�jp���Ak � �k��v��� jAkjp��Ak� �

�k���uk�� ��v���dx

����k

ZB

f�Ak � �k��uk��� f�Ak � �k��v��� �f

�Q�Ak� �

�� FUCHS et al�

�k���uk�� ��v�� dx

�� Ik � IIk � IIIk�

�������������������������������������������������������������������������

Ik � ���k

ZBr

�Z�

d�

ds�jAk � �k��v� � s�k���uk�� ��v��jp ��� s� ds dx

� pZBr

�Z�

jAk � �k��v � s��k���uk�� ��v��jp�� ��� s� ds

j��uk�� ��v�j� dx

� c�

ZBr

�jAk � �k��v�jp�� � �p��k

j��uk�� ��v�jp���j��uk�� ��v�j� dx

� c��

ZBr

�j��uk�� ��v�j� � �p��k j��uk�� ��v�jp� dx

������

For the last estimate we use the local boundedness of ��v� so that c�� �jAk � �k��v�jp�� � c�� for k large enough� Analogous calculations to thoseused in estimating the term I�k in Lemma ����� �cf� ����� show

limk��

IIk � �������

Finally by convexity of f

IIIk � ���k

ZBr

� �f�Q

�Ak � �k��v�� �f

�Q�Ak�

�� �k���uk�� ��v�� dx

�ZBr

�Z�

��f

�Q��Ak � s�k��v�����v�� ��uk�� ��v�� ds dx�

and the last term vanishes as k � �recall ��v� � L��Br��� Combining thiswith ������ and ������� inequality ������ follows and the claim of Lemma �����

Variational Methods ��

is then a consequence of ��������

In order to complete the proof of the Main Theorem we need to establish that�for the family of functionals fJ�g�������� with J��v� ��

R�j��v�jp � f����v��dx�

where f��Q��� ��� � jQj����� the quantity � occuring in Lemma ����� can bechosen uniformly in �� Precisely we have

LEMMA ����� Let A be a non zero symmetric matrix with norm ��� and letu be a local minimizer for J� in K � Then there exists a constant c� dependingonly on A such that for every t ���� �� there exists � depending on t and A suchthat the conditions

j���u��x��R � Aj � �������

E�u�BR�x��� � �������

for some ball BR�x�� � � together imply the estimate

E�u�BtR�x��� � c�t�E�u�BR�x���������

Proof� If the proposition were false we could �nd� for any c� � andt ���� ��� a sequence f�kg ���� �� and corresponding local J��minimizers fvkgand balls fBRk�xk�g � � such that conditions �������� � held� If �� ��lim infk��

�k � we have an immediate contradiction �set � � min��������

��t� f�� A��

where �� � infk�k� in ����� �here ��t� f�� A� is given by Lemma ������� since

�� � by assumption� � ��� Hence� after passing to an appropriate subse�quence� we may assume �k ��

As in the proof of Lemma ����� we de�ne the rescaled functions fukg anddeduce the convergences given in ������������� The remainder of the proof ofLemma ����� is directly analogous to that of Lemma ������ given that we canestablish the analogues of Lemma ����� �the blow�up equation� and Lemma����� �desired strong convergences�� In order to establish the blow�up equation�

� � FUCHS et al�

we again argue as in Lemma ������ cf� �Se���� Lemma �� We need to discussthe integral IIk with f replaced by f�k � i�e�

IIk � ���k

ZB�

����k � j�k��uk� � Akj�������k��uk� � Ak��

���k � jAkj�����Ak� � ���� dx

� I�k � I�k �

where

I�k �� ���k

ZB�

�k��uk�

�kdz � ���� dz

and

I�k �� ���k

ZB�

��

�k� �

�k���k��uk� � Ak� � ���� dz

for

�k �� ���k � jAkj����� �k �� ���k � j�k��uk� � Akj�����

It is immediate that I�k �jA�j

RB�

��v� � ���� dz� Further� we see

I�k � ���k

ZB�

���k � ��

k�

�k�k��k � �k���k��uk� � Ak� � ���� dz

� ���k

ZB�

�����uk� � Ak � ��kj��uk�j����k��uk� � Ak�

�k�k��k � �k�� ���� dz�

Since

��� �k��uk� � Ak

�k�k��k � �k�

��� � ���k jA�j�� � ��� as k �

Variational Methods � �

and�k��uk� � Ak

�k�k��k � �k� A�

�jA�j� pointwise a�e� by ������� we have

limk��

IIk �ZB�

h��v�jA�j � ���v� � A��

A�

jA�j�i� ���� dz�

thus the analogue of ������ is

�����������������

ZB�

pjA�jp�����v� � �p� ��jA�j���A� � ��v��A�� � ���� dz �

ZB�

��A� � ��v��A�

jA�j� ���v�

jA�j � � ���� dz����� �

By the above remarks this completes the proof of Lemma ����� modulo thestrong convergences to be shown next�

We note that analogous results hold for other choices of the f�� for examplef��Q� �� ��� � jQj��q�� � � q � p�

In order to obtain the appropriate version of Lemma ����� we let

f�k�Q� �� ���k �jAk � �kQjp � jAkjp � p �kjAkjp��Ak � Q� � ���k f�k�Q�������

where f�k�Q� ��q��k � jAk � �kQj� �

q��k � jAkj� � �kQ �

Akq��k � jAkj�

notice Lip � f�k� � ��

We now claim that estimate ����� ist still valid where c� remains independentof �k� To see this we have to establish ����� for the function f�k �

� � FUCHS et al�

Let us �rst consider the case �kjQj � ��� where � � jAj

�as in Lemma ������

Then

�������������������������������������������������

���k f�k�Q� � ���k

�Z�

�� f�k�Q�

�Ak � s�kQ���kQ� �kQ���� s� ds

� c�jQj��Z

���k � j�ksQ� Akj����� ds

� c�jQj� �

jAkj � �kjQj

� c�jQj� ��

� c����jQj��

������

Now let �kjQj � �

�� we have

�����������������

���k f�k�Q� � ���k �� Lip � f�k��kjQj�

� � jQj�k

� c����jQj�

������

Estimates ������ and ������ together with ���� � give ����� for a suitable con�stant c��

The integral IIIk from the proof of Lemma ����� now reads

���������������������������

IIIk �� ���k

ZBr

f�k�Ak � �k��uk��� f�k�Ak � �k��v��� � f�k�Q

�Ak� �

�k���uk�� ��v�� dx

� ���k

ZBr

�� f�k�Q

�Ak � �k��v��� � f�k�Q

�Ak��� �k���uk�� ��v�� dx�

������

Variational Methods � �

Since ��v� � L��Br� we have jAk��k��v�j � �

�in Br for k large enough� This

gives

���kh� f�k�Q

�Ak � �k��v��� � f�k�Q

�Ak�i�k��

��f��Q�

�A�����v�� ��������

uniformly in Br� where f��Q� � jQj�

On the other hand we already know ��uk� � ��v� in L��Br� so that ������ and������ imply�

limk��

IIIk � ��

Let us now suppose that u � W �p ���R

n� is the minimizer of I�v� �R�j��v�jp�

j��v�j dx in the class K de�ned by Dirichlet�boundary conditions and the re�quirement div v � �� We set �� ��

f��Q� �� ��� � jQj����� Q � M �

which is a smooth convex function M ����� satisfying the growth condition����� for q � ��

LEMMA ����� Let u� denote the minimizer of v R�j��v�jp� f����v�� dx in

the class K � Then u� u strongly in W �p ���R

n� as � � ��

Proof� From Korn�s inequality and

Z�

j��u��jp dx � I��u�� � I��u�� � I��u��

we infer

sup���

ku�kW �p��

� � FUCHS et al�

so that u� � u weakly in W �p as � � � for some element u � W �

p at least fora subsequence�

Using

I�u�� � I��u�� � I��u�

we get

I� u� � lim inf� �

I�u�� � lim inf� �

I��u� � I�u��������

Since I is strictly convex� we conclude u � u and we have

u� � u in W �p as � � �

�not only for a subsequence��

We now claim

lim sup� �

k��u��kp � k��u�kp������

which completes the proof using uniform convexity of Lp�

from ������ we have

I�u� � lim inf� �

I�u�� � lim inf� �

I��u�� � lim sup� �

I��u�� � lim sup� �

I��u�

that is

lim� �

I��u�� � I�u�������

Variational Methods �

Now

�������������������������

Z�

j��u��jp dx � I��u��� I�u� �Z�

�j��u�jp � j��u�j�dx

�Z�

q�� � j��u��j� dx

� I��u��� I�u� �Z�

j��u�jp dx�Z�

j��u�j dx�Z�

j��u��j dx�

�����

Notice �by lower semicontinuity�

lim sup� �

n Z�

j��u�jdx�Z�

j��u��jdxo� ����� �

and ������ is a consequence of ������ ����� and ���� ���

We de�ne �� ��nx � �� x is a Lebesgue point for ��u��

j��u��x�j � and limr��

E�u�Br�x�� � �o�

Choose x� � �� and let A �� ��u��x�� � �� � �� ��jAj� We �x a radius R such

that

j���u��x��R � Aj � �

�� E�u�BR�x��� � ��t�������

with ��t� � ���t� ���t� de�ned in Lemma ������ to be speci�ed later� Here t ischosen such that c�t

� � ��� For � small enough �recall Lemma ����� we deduce

from ������

������� j���u���x��R � Aj � ��� E�u�� BR�x��� � ��t��

hence �via Lemma ������

� FUCHS et al�

������ E�u�� BtR�x��� � ��E�u�� BR�x����

Combining �������� ������ we get E�u�� BtR�x��� ��� ��t� and

j���u���x��tR � Aj � �

�� t�nE�u�� BR�x���

�� � �

provided t�n ��t��� � ��� In this case Lemma ����� gives

E�u�� Bt�R�x��� � �

�E�u�� BR�x����

In order to iterate this argument we observe

j���u���tk��R � Aj � j���u���x��R � Aj�kXl��

t�nE�u�� BtlR�x������

Then it is easy to check that for ��t� satisfying

t�n�Xl��

��l�q ��t� �

we end up with

E�u�� BtkR�x��� � ��kE�u�� BR�x���

being valid for all k and small enough �� Now from standard arguments itfollows that there exists � � ��� �� such that

��u�� � C����BR��x��������

uniformly with respect to �� �Obviously a slight re�nement of the above calcu�lations gives ����� with � replaced by an arbitrary exponent ��� From Lemma���� we immediately deduce ��u� � C����BR��x��� which proves the �rst partof the Main Theorem� To conclude the proof we �x a compact subregion G of�� and observe the equation

ZG

���u� � ���� dx �ZG

�F � ���� dx�����

Variational Methods �

for all � � C�� �G�Rn�� div � � �� Here we have set

� � pj��u�jp��� F ���u�

j��u�j�

Suppose BR��� � G and let w denote the minimizer ofR

BR���

j��w�j� dx for

boundary values u and subject to the constraint div w � ��

Inserting � � u� w into ����� and observing that

j��x�� ����j � cR�� jF �x�� F ���j � cR�

for x � BR��� we deduce �note ��u� � L��G��

ZBR���

j��u�� ��w�j� dx � cRn���

on account of Korn�s inequality this yields

ZBR���

jru�rwj� dx � cRn���

Finally we make use of the Campanato estimate for w to get for r � R

ZBr���

jru� �ru�rj� dx � c

������r

R

�n� ZBR���

jru� �ru�Rj� dx�Rn��

��� �

hence ru � C����G�� and the proof of the Main Theorem is complete��

In contrast to the Bingham �uid model we obtain partial regularity up to aclosed set of vanishing Lebesgue measure in � for the Powell�Eyring modeland certain types of power law �uids�

� � FUCHS et al�

Let us �rst discuss the power law

W �E� � ��jEj� � ��

���������������

�� � jEj��p�� � �� p � �

or

jEjp � p �

������������������

with constants ��� �� � �� �� � �� �� We de�ne m � in such a way�depending on the various choices for ��� �� and p� that

J�u� �Z�

�W ���u��� g � u� dx min�����

is uniquely solvable in K � fv � W �m���R

n� � div v � �� v � v� on ��g� Fortechnical simplicity we just let g � �� the next results �with obvious restrictionson the exponent �� remain valid for volume forces in some Morrey space�

THEOREM ����� Let u � K denote the solution of ������ with W denedin ������� Suppose further that p � ��

a� If �� �� then there exists an open subset �� of � such that j����j � �and u � C�������R

n� for any � � ��

b� In case �� � � we have�

�� Let W �E� � ���� � jEj��p�� Then the result of a� holds�

�� Let W �E� � ��jEjp� Then there is an open subset �� of � �fx � � � x is a Lebesgue point of ��u� and ��u��x� � �g such thatj� � ��j � � and u � C�������R

n��

THEOREM ����� Let u � K denote the solution of ������ with W from������� let �� � and � � p � ��

a� Suppose that W �E� � ��jEj������� jEj��p�� Then we have the partialregularity result from Theorem ����� a��

b� If W �E� � ��jEj� � ��jEjp� then the statement from Theorem ����� b��� holds�

Variational Methods � �

REMARK ����� We conjecture that for any dissipative potential

W �E� � ��jEj� � ��

���������������

�� � jEj��p�� � �� p � �

or

jEjp� p �

��������������local� minimizers are of class C� without interior singular points�

REMARK ����� In order to get better regularity results it will be necessa�ry to study the smoothness properties of local minimizers of

R�j��v�jmdx for

growth rates m � and with respect to the constraint div v � ��

Next we consider the variational problem ����� for the dissipative potential

W �E� � ��jEj� � ��

Z jEj

�ar sinh t dt

assuming that g � L��n������Rn� for some � � � � ��

THEOREM ����� With notation introduced above let u � fv � W �� ���R

n� �div v � �� v � v� on ��g denote the unique J minimizer� Then there is anopen set �� such that j�� ��j � � and u � C�������R

n�� � � ��

Proof of Theorem ������ ������ ����� Most of the results follow from the proofof Theorem ������ In the situation of Theorem ���� we apply Theorem �����with p � � and f�E� � ��

R jEj� ar sinh t dt� It is then easy to check that the

blow�up lemma ����� is true without assuming that the matrix A is di�erentfrom zero� We therefore deduce H&older continuity of ��u� near a point x� � �if and only if x� is a Lebesgue point for ��u� andlimr��

�RBr�x��

j��u�� ���u��x��rj�dx � �� Exactly the same argument �we now let

f�E� � ���� � jEj��p� in Theorem ������ proves Theorem ����� a�� For obtai�ning the results of Theorem ����� a� and b� �� we again go through the proofof Theorem ����� taking care of the fact that now the dissipative potentials Wunder consideration are nondegenerate� hence the requirement A � � of Lem�ma ����� is super�uous� which proves the results� The statement of Theorem

��� FUCHS et al�

����� b� �� is obvious� In the situation of Theorem ����� b� we would like toapply Theorem ����� again with the choice f�E� � jEjp� Since f is not of classC� we have to consider the approximation �� � jEj��p�� � �� and to proveuniform partial regularity with respect to �� But this is just another versionof the Main Theorem ����� which does not require additional arguments�

REMARK ����� In section � we will show how to improve the above resultsin the twodimensional case�

REMARK ����� It should be clear without further comments that all theo�rems of this section extend to local minimizers�

Variational Methods ���

��� Local boundedness of the strain velocity

In this section we discuss dissipative potentials of the typeW �E� � �

�jEj� � S�E� with S � M R de�ned according to

S�E� �

���������������������������

jEj or

Z jEj

�ar sinh t dt or

jEjp� � � p � �� or

�� � jEj��p�� � � p � �� � ��

�������������������������� E � M ������

in particular� the next results apply to the classical Bingham �uid� the Powell�Eyring model and certain power laws� The reader should note that the caseW �E� � �

mjEjm � S�E�� m �� is still open�

In order to formulate our results not only for the speci�c �uid models mentio�ned above we assume

W �E� ��

�jEj� � f��jEj������

with f� � ����� R convex and nondecreasing� For v� � W �� ���R

n� withdiv v� � � we let as usual K � fu � W �

� ���Rn� � div u � �� u � v� on ��g

and consider the variational integral

J � K R� J�u� �Z�

W ���u�� dx�

the case of non�zero volume forces requires some minor modi�cations�

THEOREM ����� With W dened according to ����� let u � K denote theunique J minimizer in K � Assume further that

limt�� t��f��t� � �������

Then u is H�older continuous on � with any exponent � �� Moreover� theclassical derivative ru�x� exists for Lebesgue almost all x in ��

��� FUCHS et al�

REMARK ����� Condition ����� holds for f��jEj� � S�E� with S de�nedaccording to ������

REMARK ����� We do not claim that the set of points x for which ru�x�exists is open�

REMARK ����� The statement of Theorem ����� is true for any continuousfunction f� satisfying ����� and for any local J�minimizer in K � But in thismore general setting the existence of minimizers is not guaranteed� we maythen look at the relaxed variational problem and according to �Fu��� �FS�� thestatement of Theorem ����� extends to minimizers of the relaxed problem�

Of course ����� excludes quadratic growth at in�nity but it is possible to replace����� by the following assumptions�

f� is of class C� on some interval �a���� a �������

� limt��

tf ���t� exists in ����������

THEOREM ����� Dene W according to ����� and let ������ ����� hold forf�� Then the conclusion of Theorem ��� is still true�

REMARK ����� Conditions ������ ���� are easily checked forf��jEj� � S�E� with S from ������

Next we state the main result of this section�

THEOREM ����� Let u � K denote the unique J minimizer for the dissi�pative potential W from ������ Let ����� hold and assume further�

supt�a

tf ���t� ������

there exists L � a such that sups�L

s

js� tj

�����f���s�

s� f ���t�

t

����� � as t����� �

Then ��u� is a locally bounded function�

Variational Methods ���

REMARK ����� From ��u� � L�loc��� M � we deduce with Korn�s inequalitythat� for any p ��� we have u � W �

p�loc���Rn�� This provides an alternative

proof of the H&older results from Theorem ����� and ������

REMARK ����� In the papers �Fu�� and �FS�� we proved that the nonli�nearities S occuring in ����� satisfy all the assumptions from Theorem ������in particular� we get local boundedness of the strain velocity for the classicalBingham �uid�

REMARK ���� The statement of Theorem ����� extends to the followingsetting� with W as in the theorem we let W � � W � � where � � M R

is just continuous with compact support� It was then shown in �FS�� thatthe quasiconvex envelope QW � of W � equals W outside a large ball in M �moreover� we have ��u� � L�loc��� M �� where u � K now denotes a minimizerof the relaxed problem

R�QW ����u�� dx min in K �

We now will give the proofs of the various regularity results�

For Theorem ����� we follow the paper �Fu�� replacing the exponent p occu�ring there by � and taking ��u� in place of ru� Then H&older continuity of u isproved as in �Fu��� Theorem �� With the same changes di�erentiability almosteverywhere of minimizers is a consequence of �Fu��� Theorem �� We leave thedetails to the reader�

Proof of Theorem ������ We just show di�erentiability almost everywhe�re� We may assume that f� is of class C� on ����� satisfying in addition� � �

tf ���t� � A� on ����� for some A� � �� if this is not the case then we

may either proceed as in the proof of Theorem ����� by splitting the integralsinvolving f �� or we may argue as in �FS�� by writing W � �

�j � j� � f��j � j� � �

where f� � ����� R is a convex� nondecreasing C� function equal to f� on�a��� and � denotes a function with compact support� Exactly this situationwas studied in �FS��� Theorem ���� The main step towards the desired claimis

LEMMA ����� Let G � fx � � � supr��

�RBr�x�

jruj�dz ��g� Then� for x � G�

there exist positive constants Kx and Rx � ��dist �x� ��� with the following

property� if� for some R � Rx� we have oscBR�x�u � KxR� then oscBR���x�u ���oscBR�x�u�

��� FUCHS et al�

Having proved Lemma ����� we may argue as in �Fu�� to get the result ofTheorem ������

Proof of Lemma ������ Assuming that the statement is wrong we �nd a pointx � G �w�l�o�g� x � �� and sequences Kk �� Rk � such that

�k �� oscBRku � KkRk� oscBRk��u ��

��k������

Let us de�ne uk�z� � ��k�u�Rkz� � �u�Rk�� z � B�� �u�Rk �

R�BRk

u dx� From

����� we deduce

oscB�uk � �� oscB���uk

�� �uk�� � �������

Clearly div uk � � and uk locally minimizes the energy

Jk�v� �ZB�

�j��v�j� � fk���v�� dx

with respect to the constraint div v � �� Here we have abbreviated

fk�E� � R�k�

��k f��

�kRkjEj��

From our modi�ed assumptions concerning f� we deduce f��t� � A�

�t� � f�����

Using Rk���k � as k � we see that the integrand of the energy Jk satis�es

the hypotheses of �Gi�� chapter V� Theorem ���� hence

ZBR���x�

j��uk�j�dz � c��R��

ZBR�x�

juk � �uk�Rj�dz �Rn�������

for arbitrary balls BR�x� � B�� Here and in what follows c�� c� � � � denotevarious positive constants independent of k� �Remark� in order to obtain theCaccioppoli inequality ������ for the symmetric derivative and also under the

Variational Methods ��

constraint div u � � one has to modify Giaquinta�s argument along the linesof �GM�� see also proof of Lemma ����i���

Combining ����� and ������ we get boundedness of the sequence f��uk�g in thespace L�

loc�B�� M � and by Korn�s inequality fukg is bounded in W ���loc�B��R

n��After passing to a subsequence we may therefore assume

���������������

uk � u weakly in W ���loc�B��R

n��

uk u strongly in L�loc�B��R

n�� and a�e�

uk � u weakly in L�loc�B��R

n�

������

for some function u � W ���loc�B��R

n�� In a next step we prove a uniform localH&older condition for the functions uk� To this purpose �x a ball BR�x� � B����and let vk denote the solution of the Stokes problem

���������������������

ZBR�x�

��vk� � ���� dz � � for � � �W �

��BR�x��Rn�� div � � ��

div vk � � a�e��

vk � uk on �BR�x��

������

For vk we have the estimate

ZBr�x�

jrvkj� dz � c��r

R�n

ZBR�x�

jrvkj� dz� � � r � R�������

with c� only depending on the dimension n� ������ clearly implies

ZBr�x�

jrukj� dz � c�f� rR�n

ZBR�x�

jrukj� dz �Z

BR�x�

jruk �rvkj� dzg�������

�� FUCHS et al�

It therefore remains to discussR

BR�x�

jruk � rvkj� dz or equivalently �obser�

ve Korn�s inequality� the termR

BR�x�

j��uk� � ��vk�j� dz� From the minimum

property of uk we deduce

�����������

ZBR�x�

��uk� � ���� �Dfk���uk�� � ���� dz � � for all

� � �W �

��BR�x��Rn�� div � � ��

�����

with Dfk���uk�� � Rk���k f ����kR

��k j��uk�j�j��uk�j����uk�� Let l �� lim

y���yf ���y�

andsk �� �kR

��k j��uk�j� Then

ZBR�x�

�� � l�j��uk�� ��vk�j� dz �������

ZBR�x�

�� � l� ��uk� � ���uk�� ��vk�� dz �

ZBR�x�

�� ��

skf ���sk�� ��uk� � ���uk�� ��vk�� dz�

ZBR�x�

�l � �

skf ���sk����uk� � ���uk�� ��vk�� dz

������

ZBR�x�

jl � �

skf ���sk�j j��uk�� ��vk�j j��uk�j dz�

hence

ZBR�x�

j��uk����vk�j� dz � c�

ZBR�x�

jl� �

skf ���sk�j j��uk�j j��uk����vk�j dz������

Variational Methods ��

Given a large positive number � we select y� � such that

jl � �

yf ���y�j � ��� for all y � y������ �

Therefore� if y is some number � �� we can �nd k� � k���� such that

���� �� jl � Rk���k y��f ����kR

��k y�j � ���

holds for all k � k�� Note that k� does not depend on y�Let us introduce the sets �k �� fz � BR�x� � j��uk��z�j � �g and ��k �BR�x�� �k� Then

Z�k

jl � �

skf ���sk�j j��uk�j j��uk�� ��vk�j dz �

�A��Z

BR�x�

j��uk�� ��vk�j dz

and Z��k

jl � �

skf ���sk�j j��uk�j j��uk�� ��vk�j dz � ���� �� �

��

ZBR�x�

j��uk�j j��uk�� ��vk�j dz�

Applying Young�s inequality we derive from �����

ZBR�x�

j��uk�� ��vk�j� dz � c�fRn�� � �� ��

ZBR�x�

j��uk�j� dzg�������

Inserting ������ into ������ we end up with

ZBr�x�

jrukj� dz � c f�� rR�n �

��Z

BR�x�

jrukj� dz � �� � ��Rng������

��� FUCHS et al�

being valid for all r � R and all k � k����� Going through the proof ofLemma ��� in �Gi�� chapter III� we deduce from ������ the growth estimate ��su�ciently large� k � k�����

ZBr�x�

jrukj� dz � c��r

R�n��

ZBR�x�

jrukj� dz

which implies uk � C�����B��Rn� with local H&older constant independent

of k� especially uk u locally uniformly on B� which gives �recall ������oscB���

u � ���� On the other hand we have for t � �

ZBt

jrukj� dz � R��nk ���k

ZBtRk

jruj� dx � R�k�

��k

Z�BRk

jruj� dx �

as k � so that r u � � which is the desired contradiction��

Proof of Theorem ������ We �rst derive a variational inequality for the mi�nimizer u� Consider � � C�

����Rn�� div � � �� and � �� For t � we

have Z�

W ���u�� dx �Z�

W ���u� � t����� dx

which implies ��j��u�j a� �� � fx � � � jEu�x�j a� �g� etc� �

�t

Z�

�j��u� � t����j� � �

�j��u�j�dx�

��t

Z�j��u�j�a��

f��j��u� � t����j�� f��j��u�j� dx

� �t

Z�j��u�j�a��

f��j��u�j�� f��j��u� � t����j� dx�

Let us further assume that t � ��kr�k�� Then� on the set �j��u�j a � ��� wehave

Variational Methods ���

�t�f��j��u� � t����j�� f��j��u�j�� �

f ���j��u�j� �j��u�j ��u� � ���� as t � ��

on �j��u�j � a� �� we observe �in case � � a���

t�f��j��u�j�� f��j��u� � t����j�� � �M j����j�

where M denotes the Lipschitz constant for f� on ��� �a�� After passing to thelimit t � � we thus get

Z�

��u� � ���� dx�Z

�j��u�j�a��f ���j��u�j�

j��u�j��u� � ���� dx �

Z�j��u�j�a��

��M�j����j dx�

Next we pass to the limit � � �� replace � by �� and use an approximationargument to derive

�����������������

Z�

��u� � ���� dx�Z

�j��u�j�a�f ���j��u�j�

j��u�j��u� � E�dx �

MZ

�j��u�j�a�j����jdx� � � �

W �����R

n�� div � � ��

������

We �x a ball BR�x�� and denote by v the unique solution of the Stokes problem

�������Z

BR�x��

��v� � ���� dx � � for all � � �W �

��BR�x���Rn�� div � � ��

v � u on �BR�x��� div v � � a�e� on BR�x���

������

According to Lemma ���� v satis�es

ZBr�x��

j��v�� ���v��x��rj�dx � c��r

R�n�

ZBR�x��

j��v�� ���v��x��Rj�dx�������

��� FUCHS et al�

� � r � R� for some constant c� � c��n�� From ������ we deduce

���������������

ZBr�x��

j��u�� ���u��x��rj�dx � c�fZ

BR�x��

j��u�� ��v�j�dx

�� rR�n�

ZBR�x��

j��u�� ���u��x��Rj�dxg�������

In order to estimate the �rst integral on the right�hand side of ������ we in�

troduce the quantities s � j��u�j� t �� R�BR�x��

j��u�j�dx���

� we further observe

that ������ remains valid with a replaced by the number L de�ned in ��� ��provided M now denotes the Lipschitz constant for f� on ��� �L�� Let us alsoassume that

t �L�������

From ������ and ������ we then deduce by letting � � u� v�

�� ��

tf ���t��

ZBR�x��

j��u�� ��v�j�dx �

�� ��

tf ���t��

ZBR�x��

��u� � ���u�� ��v�� dx �

ZBR�x��

��u� � ���u�� ��v�� dx�

ZBR�x����s�L�

f ���s��

s��u� � ���u�� ��v�� dx�

ZBR�x��

tf ���t���u� � ���u�� ��v�� dx�

ZBR�x����s�L�

f ���s��

s��u� � ���u�� ��v�� dx �

Variational Methods ���

MZ

BR�x����s�L�j��u�� ��v��j dx�

ZBR�x����s�L�

��

tf ���t��

sf ���s����u� � ���u�� ��v�� dx�

ZBR�x����s�L�

tf ���t���u� � ���u�� ��v�� dx � ���� �

�M � L supy��L

f�yf ���y�g�

ZBR�x����s�L�

j��u�� ��v�j dx�

ZBR�x����s�L�

sj�tf ���t��

sf ���s�j j��u�� ��v�j dx �

c�

ZBR�x����s�L�

j��u�� ��v�j dx� ��t�Z

BR�x����s�L�js� tj j��u�� ��v�j dx

where ��t� � sups�L

ns

js�tj jf ���s�s� f ���t�

tjo�

With Young�s inequality we get

ZBR�x��

j��u�� ��v�j�dx � c�fjBR�x�� � �s � L�j

���t��Z

BR�x��

js� tj�dxg �

c�fjBR�x�� � �s � L�j� ��t��Z

BR�x��

j��u�� ���u��x��Rj�dxg�

For the measure of BR�x�� � �s � L� we observe �recall �������

jBR�x�� � �s � L�j � �t

ZBR�x����s�L�

js� tj dx �

c �tjBR�x�� � �s � L�j���

ZBR�x��

j��u�� ���u��x��Rj�dx����

��� FUCHS et al�

hence ZBR�x��

j��u�� ��v�j�dx � c���t�Z

BR�x��

j��u�� ���u���x��Rj�dx�

��t� � �t�� ��t���

Inserting this into ������ we �nally have shown

ZBr�x��

j��u�� ���u��x��rj�dx � c�f� rR�n� � ��t�g

ZBR�x��

j��u�� ���u��x��Rj�dx������

Note that ����� is valid for any � � r � R � dist �x�� ���� x� � �� provided������ holds� It is now more or less standard to derive local boundedness of��u� from ����� �see �Se����� For completeness we give the arguments� let

#�x�� R� � �Z�

BR�x��

j��u�� ���u��x��Rj�dx����

����� implies

#�x�� r� � c�f rR

� ��t��r

R��n�g#�x�� R������

for any r ���� R� with ��s� � as s � ���s� �q��s��� and ����� is

valid under the assumption that �R�

B�R�x��

j��u�j�dx��� �L� We �rst select

� ���� �� according to

�c�p� � ������ �

The number L is de�ned through the condition

��n���s� � � for all s � L�������

Let M � maxf�L� Lg � �� We then consider a point x� � � for whichlim���

R�B��x��

j��u�j�dx exists �which is true for a�a� x� � ���

Variational Methods ���

Two cases can occur�

Case �� lim���

Z�

B��x��

j��u�j�dx �M�

Case �� lim���

Z�

B��x��

j��u�j�dx M��

In case � we let

R� � sup fr ���� dist �x�� �����Z�B��x��

j��u�j�dx M� for all � � � rg�

Then

� Z�

Br�x��

j��u�j�dx���

�L������

and �observe �������

��n����

Z�

Br�x��

j��u�j�dx����� �������

for all � � r � R�� By ������ ���� �� ������ and ������ we �nd

#�x�� �R� �p�#�x�� R�� � � R � R��

which gives after iteration �starting at R � R��

#�x�� �kR�� � �k�#�x�� R��� k � N �������

��� FUCHS et al�

From ������ we deduce

���� Z�

B�kR�

�x��

j��u�j�dx��� � � Z

�BR� �x��

j��u�j�dx������ �

k��Pi��

���� Z�

B�i��R��x��

j��u�j�dx��� � � Z

�B�iR�

�x��

j��u�j�dx������ �

k��Pi��

�� � ��n���#�x�� �iR�� �

k��Pi��

�� � ��n���� i�#�x�� R�� � c������ Z

�BR� �x��

j��u�j�dx���

c����� ��

��p� �� � ��n����

and therefore

lim���

� Z�B��x��

j��u�j�dx��� �

�� � c������� Z�BR� �x��

j��u�j�dx���

In case R� � dist �x�� ��� we get

lim���

� Z�B��x��

j��u�j�dx��� �

�� � c������jB����j��� dist �x�� ����n��Z

�j��u�j�dx

����

for R� � dist �x�� ��� we just use

� Z�

BR� �x��

j��u�j�dx��� �M�

Collecting our results we have �nally shown

j��u��x��j � �� � c������

max fM� jB����j��� dist �x�� ����n��Z

�j��u�j� dx���g

Variational Methods ��

for almost all x� � �� this completes the proof of Theorem �������

REMARK ����� For su�ciently regular boundary data v� and �� of classC� we expect global L��bounds for the tensor ��u��

REMARK ����� We proved Theorem ����� for the case of zero volume forcesf � The statement and also the proof remain valid if we consider volume forcesf in the space L����Rn� � L��n����

loc ���Rn� for some � � � � ��

REMARK ����� As in section � all our results continue to hold for localminimizers of the energies under consideration�

�� FUCHS et al�

��� The two�dimensional case

This section continues the study of the smoothness properties of the strainvelocity which we started in section �� We will show that for two�dimensionalproblems the tensor ��v� is continuous on � without singular points providedv is the minimizer of the energy functional associated to the Bingham� thePowell�Eyring and certain power law �uid models�

Let us �rst assume that W is of the form

W �E� � ��jEj� � ��

���������������

�� � jEj��p�

orZ jEj

�ar sinh t dt

�������������� E � M ������

with constants ������ � �� � � p � �� For volume forces g in the spaceL����Rn� we consider the variational problem ����� in the class K � fv �W �

� ���Rn� � div v � �� v � v� on ��g�

THEOREM ����� Let v � K denote the solution of ������ with W denedas in ������ Then v has second generalized derivatives� i�e� v � W �

��loc���Rn��

THEOREM ����� With notation as in Theorem ����� let us in addition as�sume that g � L��n�������Rn� for some � � � � �� Then there is an opensubset �� of � such that v � C�������R

n� for any � � � and Hn�������� � ��Here Hn�� denotes the �n� �� dimensional Hausdor measure� In particular�if n � �� then we have no singular points� i�e� �� � ��

REMARK ����� Our �rst result holds for any dimension n � ��

Proof of Theorem ������ Instead of modifying the arguments used for theproof of Lemma ���� we give a variant which can be found in the paper �FS���W�l�o�g� we may assume that �� is of class C� �otherwise we replace � bya smooth subdomain�� Let u � W �

� ���Rn� denote the unique solution of the

linear Stokes problem���������������

Z�

��u� � ���� dx �Z�

g � �dx for all � � �W �

����Rn�� div � � ��

div u � � on � and

uj�� � ��

�����

Variational Methods ��

From the work �L�� we deduce that u is in the space W �� ���R

n� together withthe estimateZ

jruj� � jr�uj�dx � c����Z�

jgj�dx������

Using ����� as well as the minimum property of v we �nd

Z�

��W��

���v�� � ����� ��u� � �����dx � ������

for any � � �W �

����Rn�� div � � �� According to �L�� and �LS� there exists a

pressure function p � L���� such that ����� can be written as

Z�

��W��

���v�� � ����� ��u� � �����dx �

Z�

p div �dx����

being valid now for any � in the space�W �

����Rn�� Let � � C�

���� and considerh � Rn such that jhj � dist �spt�� ���� We further let "g�x� � g�x�h��g�x�for functions g� ���� implies

Z�

��W��

���v��x� h��� �W

�����v��x��

�� ����"hv� dx �

Z�

��"hu� � ����"hv� dx�

Z�

"hp div ���"hv� dx�

or equivalently

�������������������������������������������

Z�

��h�W��

���v��x� h��� �W

�����v��x��

i� ��"hv� dx �

��Z�

�h�W��

���v��x� h��� �W

�����v��x��

i� �"hv �r���

Z�

����"hu� � ��"hv� dx� �Z�

���"hu� � �"hv �r�� dx�

Z�

"hpr�� �"hv dx�

����

��� FUCHS et al�

Here we use the notation a� b � ���aibj � ajbi� for a� b � R

n � From de�nition����� we deduce

��jXj� � D�W �Y ��X�X� � (jXj�

for any X� Y � M with a suitable constant ( �� and we may use thisinequality to get a lower bound for the left�hand side of ����������������������������������

Z�

��j��"hv�j� dx �

c�n�Z

��j��"hv�j�dx����Z

��j��"hu�j�dx�Z�

j"hvj�jr�j�dx���

�Z�

�j��"hu�j j"hvj jr�j dx�Z�

"hpr�� �"hv dxo�

��� �

For the last integral on the right�hand side of ��� � we observeZ�

"hpr�� �"hv dx � �Z�

"hpr� � �"hv dx �

�k"hpr�k� kr��"hv�kL���� � �k"hpr�k� k���"hv�kL�����

where

k"hpr�k� � sup fZ�

"hpr� � w dx � w � �W �

����Rn�� krwkL���� � �g

� c�jhj�Z�

p� dx���

Inserting this into ��� � and using Young�s inequality we �nd thatZ�

��j��"hv�j�dx � c�

Z�

�j��"hu�j� � j"hvj� � jhj�p�

�dx

where all integrals have to be calculated w�r�t� spt �� Recalling ����� as wellas standard estimates for the pressure p we get the �nal inequalityZ

��jr���v��j�dx � c�

Z�

jgj�dx�

Variational Methods ���

and the proof of Theorem ����� is complete��

REMARK ����� With similar arguments �for details see �Se���� �Se���� wecan show the following� suppose that g � L��n�������Rn�� Then there is aconstant c � with the property

ZBR���x��

jr��v�j�dx � cnRn���� �R��

ZBR�x��

j��v�� ���v��x��Rj�dxo

for any BR�x�� � �� This estimate can be used to give a �direct� proof ofTheorem ����� following standard arguments used in the theory of nonlinearelliptic systems �see �Gi���

Proof of Theorem ������ From the proofs of Theorem ����� a� and Theorem���� we immediately deduce that v is of class C���� � � �� in a neighborhoodof x� � � if and only if

i� supr��

j���v��x��rj �� and ii� limr��

�RBr�x��

j��v�� ���v��x��rj�dx ���

According to Theorem ������ i� is true for every x� � �� By Poincar�e�s inequa�lity �and Theorem ������ we have

Z�

Br�x��

j��v�� ���v��x��rj�dx � c r�Z�

Br�x��

jr�vj�dx�

and

r�Z�

Br�x��

jr�vj�dx � as r � � for Hn�� � almost all x� � ��

We next discuss the Bingham case and give the proof of a theorem which wasobtained in �Se����

��� FUCHS et al�

THEOREM ����� Suppose that n � � and deneI�u� �

R���j��u�j� �p

�k�j��u�j � g � u� dx on the class K � fv � W �� ���R

�� �

div v � �� v � v� on ��g� � and k� denoting positive constants� Supposefurther that g � L������R�� for some � �� Then� for the unique I minimizerv � K � we have ��v� � C���� M ��

REMARK ����� We do not claim that ��v� belongs to some H&older space�

REMARK ����� We note that Theorems ������ ����� and ����� hold underthe assumption that v is only locally minimizing�

The proof of Theorem ����� is divided in several steps� �rst we show an ana�logue of Theorem ����� for minimizers of the energy I introduced above�

THEOREM ����� Suppose that n � � and g � L����Rn�� Then the uniqueI minimizer in the class K is in the space W �

��loc���Rn��

Proof of Theorem ������ We argue by approximation� Let F � M �����denote a convex Lipschitz function of class C� and de�ne

J�u� �Z�

��

�j��u�j� � F ���u��� g � u� dx� u � K �

Let v � K denote the unique J�minimizer� From the proof of Theorem �����it should be clear that v � W �

��loc���Rn� but now we need an estimate for the

second generalized derivatives of v which does not involve an upper bound forD�F � in the concrete approximation below this quantity can not be controled�We write W � �

�j � j� � F and de�ne u as in ����� assuming that �� is smooth

enough� Then as before

Z�

�W

�E���v�� � ���� dx �

Z�

��u� � ���� dx�Z�

p div �dx�����

for any � � �W �

����Rn�� Here p � L���� is a pressure function such thatR

�p dx � �� It is easy to show that

Z�

p� dx � c�

Z�

�j��v�j� � j��u�j�� dx�����

Variational Methods ���

where c� depends also on F but only the Lipschitz constant of F enters� Wereplace � in ����� by ��� for some � � �� � � � � n and get

Z�

��f�W�E

���v��g � ���� dx �Z�

����u� � ���� dx�Z�

��p div �dx�������

Consider now � � C�� ���� � � �� � � V�� and insert � � �����v��� in equation

������� We have

�����������������������������������������

R�����f�W�E ���v��g � ����v� dx �

Z�

��D�W ���v�������v�� ����v�� dx �

Z�

��j����v�j�dx�Z�

����F ���v�������v�� ����v�� dx �

Z�

��j����v�j�dx

������

by convexity and smoothness of F � On the other hand

Z�

����f�W�E

���v��g � ����v� dx �

Z�

��f�W�E

���v��g � �������v � ��� dx�

Z�

��f�W�E

���v��g � �r�� � ���v � ��� dx � ������ �

Z�

����u� � �������v � ��� dx�

Z�

��p div ������v � ��� dx�

Z�

��f�W�E

���v��g � �r�� � ���v � ��� dx �

�I� � �II�� �III��

��� FUCHS et al�

and we discuss these integrals separately� Recalling ����� we use Young�s ine�quality with arbitrary � � to see �w�l�o�g� � � � � �����������������������������

�I� � c����Z�

jr�uj�dx� �Z�

j�������v � ���j�dx

� c����fZ�

jgj�dx �Z�

jr�j�j��v � �j�dxg

��Z�

��j����v�j�dx�

������

Next we have

�II� �Z�

��pr�� � ���v � �� dx � �Z�

p ���r�� � ���v � ��� dx

�Z�

jpj j��r��j j��v � �j dx�

Z�

jpj jr��j j�����v � ��j dx�

Let us assume that � is supported on some ball BR�x�� and jrk�j � c�R�k�

k � �� �� Then� again with Young�s inequality� we deduce

�II� � c�nR��

ZBR�x��

�jpj� � j��v � �j�� dx�

�Z�

jpj jr�jp� ���j�����v � ��j dxo�

c ���R��

ZBR�x��

�jpj� � j��v � �j�� dx� �Z�

��j�����v � ��j� dx�

UsingZ�

��j�����v � ��j� dx � c�nZ�

j��������v � ���j� dx�Z�

jr�j� j��v � �j�dxo�

c�nZ�

j�������v � ���j� dx�Z�

jr�j� j��v � �j�dxo�

c�nZ�

��j����v�j� dx �Z�

jr�j� j��v � �j�dxo

Variational Methods ���

we see� combining ������� ������ and the above estimates� that �after appro�priate choice of ��

�Z

BR�x��

��j����v�j� dx �

c��n ZBR�x��

jgj�dx�R��Z

BR�x��

j��v � �j�dx�R��Z

BR�x��

jpj�dxo� �III��

Here � is some positive constant� and we have used the fact that all integralshave to be calculated with respect to the ball BR�x���In �III� we integrate by parts to get �A � M � � ��

��III� �Z�

��W�E

���v��� �W

�E�A�

�� ���r�� � ���v � ��� dx

�Z�

����W�E

���v��� �W

�E�A�

��� j�� r��j j��v � �j dx�

Z�

����W�E

���v��� �W

�E�A�

��� jr��j j�����v � ��j dx

� c�����n ZBR�x��

R��j�W�E

���v��� �W

�E�A�j�dx�

ZBR�x��

R��j��v � �j�dxo� �

ZBR�x��

j����v�j�dx�

Putting together the various estimates� we see by letting � � � on BR��x���that

���������������

ZBR���x��

j����v�j� dx � c��n ZBR�x��

jgj�dx�R��Z

BR�x��

j��v � �j�dx

�R��Z

BR�x��

jp� �p�x��Rj�dx�R��Z

BR�x��

j�W�E

���v��� �W

�E�A�j�dx

o�

������

Here we used that p can be replaced by p� �p�x��R�

��� FUCHS et al�

After these preparations we give a brief outline of the proof of Theorem ������

Let now v denote the I�minimizer in K � For notational simplicity we assume� � �

�� k� � ��

p� and de�ne

F��E� �q� � jEj�� � �� E � M �

Let v� � K denote the corresponding minimizer of J�� From Lemma ���� wehave v� v as � � � strongly in W �

� ���Rn�� and by the foregoing calculations

������ holds for the functions v� with pressure p� and W replaced by W��E� ���jEj� �F��E�� By ����� the pressure is estimated independent of �� moreover�

we see that

j�W�

�E�M�j � jM j� jM jq

� � jM j�� jM j� ��

This proves that the right�hand side of ������ is bounded independent of �which clearly implies v � W �

��loc���Rn�� Unfortunately ������ does not extend

to the limit � � � for the obvious reason that the last integral on the right�handside of ������ may have not limit as � � ��

After these preparations we give an outline of the proof of Theorem �����assuming from now on that all the hypothesis are satis�ed� As in the proof ofthe previous theorem we consider the approximations fv�g � K de�ned as theminimizers of the energy

J��u� �Z�

��j��u�j� �p�k�

q� � j��u�j� � g � u� dx�

Let BR�x�� denote a disc in �� Then� using arguments similar to the calcula�tions in the proof of Theorem ������ we get the following version of estimate������ �compare �Se���� �������

ZBR���x��

jr��v��j�dx � c�n�

� ��q

� � j���v���x��Rj���

� �R�

ZTR�x��

j��)v��j�dx�R�o�

������

Variational Methods ��

TR�x�� � BR�x�� � BR��x��� )v� � v� � ���v���x��R�x � x�� � u�� where u�denotes a rigid motion which is chosen according to

ZTR�x��

j)v�j�dx � c�R�Z

TR�x��

j��)v��j�dx�

Here and in the sequel ���x��R indicates the mean value over TR�x��� We passto the limit � � � in ������ and obtain

���������������

j���v��x��Rj�!�x�� R� � � c�n�� � j���v��x��Rj��

� �R�

ZTR�x��

j��v�� ���v��x��Rj�dx�R�j���v��x��Rj�o�

�����

!�x�� r� �R

Br�x��

jr��v�j�dx� It can be shown that the constant c� just depends

on �� k� and the L����norm of f � By Poincar�e�s inequality we have

ZTR�x��

j��v�� ���v��x��Rj�dx � c�R�Z

TR�x��

jr��v�j�dx�

inserting this into ����� we end up with

���������j���v��x��Rj�!�x�� R� � � c�

n�� � j���v��x��Rj��

��!�x�� R�� !�x��R��� �R�j���u��x��Rj�

o�

�����

In order to prove continuity of the strain velocity we will use ����� to obtainthe following preliminary result� if j���v��x��Rj is rather large� then we get agrowth estimate for !�x� r� when x is in some neighborhood of x�� The caselim sup

R��j���v��xo�Rj � � has to be discussed separately�

To be precise� let K��t� �c��t���

t�c��t���� t � ��

�� FUCHS et al�

Using Poincar�e�s inequality once more� we have

jj���v��xo�R�j � j���v��x��Rjj � c !���x�� R��

hence

jj���v��x����iRj � j���v��x��Rjj � c i��Xs��

!���x�� ��sR�

for any i � N � We �x �� ���� �� and de�ne the functions

K�t� � max fK��t���� ��g� D�t� ��

��qK�t�

� t ��

Then from estimate ����� we deduce �see �Se���� Lemma ����

LEMMA ����� Let � � and suppose that

j���v��x��Rj � ��

and

!�x�� R� ���R��

�� �� �� �

c D���

��

hold for some x� � � and R �� Then� for any i � N� we have

!�x�� ��iR� � Ki���

h!�x�� R� �

��R��

��i��Xs��

���si� Ki���

h �

c D���

i�

and

� � j���v��x����iRj � j���v��x��Rj� ��

Variational Methods ��

The proof of the lemma is obtained via induction� i�e� iteration of inequality������

Consider next x� � � with the property

lim supR��

j���v��x��Rj ������ �

Of course ���� � implies the existence of � � and some small radius R �such that

�������j���v��x��Rj �� �

!�x�� R� ���R���

���� ��

�c�D���

���

������

By continuity we �nd a disc B���x�� such that inequalities ������ hold with x�replaced by x for any x � B���x��� from Lemma ����� we see

!�x� ��iR� � Ki���� �

c D���

��� x � B���x��� i � N �

so that �by the Dirichlet�growth theorem� ��v� is H&older continuous in a neigh�borhood of x��

So� if condition ���� � holds� then ��v� is continuous in some open neighborhoodof x�� lim

R�����v��x��R � ��v��x�� exists and j��v��x��j � �� Suppose now

lim supR��

j���v��x��Rj � ��������

First we note that� for any x � �� we have the pointwise de�nition

��v��x� � limR��

���v��x��R�

Namely� if ���� � holds at x � �� then the existence of the limit was provedabove� in case that ������ holds for x we clearly have lim

R�����v��x�R � �� The�

refore ������ can be written as ��v��x�� � ��

��� FUCHS et al�

Let � denote some number �� We �x some subdomain �� b � of � andcalculate a small radius R��� such that

!�x�R� ���R��

�� �� �� �

c D���

��������

holds for any x � �� and R � R���� From ������ we deduce the existence ofR� ���� R���� with the property

j���v��x��R�j � ���

hence we �nd � � such that

j���v��x�R�j � ��������

for any x � B���x��� We claim

j��v��x�� ��v��x��j � j��v��x�j � ��������

on B���x��� So consider x � B��

�x��� If j���v��x��j � �� for any ���� R��� thenj��v��x�j � �� and ������ holds� Otherwise we �nd some R� ���� R�� such thatj���v��x�R�j ��� From ������ we then deduce the existence of R� ��R�� R��such that

j���v��x�Rj � ���������

By Lemma ����� � with x� � x and R � R� � we get

j���v��x���iRj � j���v��x�Rj� ��

Passing to the limit i � and using ������� we get j��v��x�j � ��� hence������ is established� Since � was arbitrary� continuity of ��v� at x� followsand the proof of Theorem ����� is complete�

REMARK ����� Having proved continuity of ��v� we now can state that�j��v�j �� � fx � � � j��v��x�j �g is open and we may therefore de�nethe �separating line� ��j��v�j �� � � � L and ask if L is a smooth curveseparating the rigid zone ���v� � �� from the region ���v� � ���

Variational Methods ���

��� The Bingham variational inequality in di�

mensions two and three

Up to now our discussion of the quasi�static �ow for generalized Newtoni�an �uids was limited to the variational setting starting from the equation ofmotion ������ in which the nonlinearity �rv�v � ��iv

jvi� is neglected� In thissection we are going to investigate the �realistic case� given by equation �������As already mentioned we are then confronted with the same di�culties as incase of the nonlinear stationary Navier�Stokes system which forces us to workin dimension � or �� Most of the material presented here is taken from thepaper �FS��� For the readers convenience we start with some known existenceresults concerning homogeneous boundary data�

THEOREM ����� Suppose that n � � or �� let f � L����Rn� be given anddene

W �E� � ��jEj� � ��

�������������������

�� � jEj��p�

or � E � M �Z jEj

�ar sinh t dt

������������������ E � M �

��� ��� � �� � � p � �� Then there exists a function

v � V� � fu � �W �

����Rn� � div u � �g such that

Z�

f�W�E

���v�� � ���� � �rv�v � �g dx �Z�

f � �dx�����

for every � in V��

REMARK ����� As in �Ga�� or �L�� we obtain uniqueness if the L��norm ofthe volume force term f is su�ciently small�

Proof of Theorem ������ Consider the operator T � V� � w v � V�� vdenoting the unique solution of

Z�

�W

�E���v�� � ���� dx �

Z�

f � �dx�Z�

�rw�w � �dx�

��� FUCHS et al�

� � V�� which for example is obtained by minimizing

V� � u Z�

fW ���u�� � �rw�w � u� f � ug dx�

It is an easy exercise to show conpactness of T � moreover� the hypothesis ofthe Leray�Schauder �xed point theorem hold� let w � �Tw for some w � V�and � � � � �� Then we getZ

�W

�E��

���w�� � ���� dx �

Z�

f � �dx�Z�

�rw�w � �dx�

hence Z�

�W

�E��

���w�� � ��w� dx �

Z�

f � w dx�

Using ellipticity we �nd that

���

Z�

j��w�j�dx � kfkL�kwkL��

and in conclusionZ�

j��w�j�dx � cZ�

jf j�dx

for a positive constant independent of ��Clearly� any function v � V� with Tv � v� is a solution of ������

THEOREM ����� �the homogeneous Bingham variational inequality�

For n � � or � and f � L����Rn� there exists at least one function v � V�satisfying the homogeneous �BVI����������������������

Z�

����v� � ���w�� ��v�� �p�k��j��w�j � j��v�j��

�rv�v � �w � v� dx �Z�

f � �w � v� dx for any w � V��

�����

Here �� k� denote arbitrary positive constants�

Variational Methods ���

REMARK ����� Uniqueness for ����� holds if kfkL� is small enough�

Proof of Theorem ������ �see �DL�� For notational simplicity we let � � ���

and k� � ��p�� With � � de�ne W��E� � �

�jEj� �

q� � jEj�� E � M �

and let v� � V� denote a solution of ����� with W � W�� We show that� afterpassing to a subsequence� we have v� v as � � � for some function v � V�which is a solution of �BVI�� Consider w � V�� From ����� we get

Z�

�W�

�E���v��� � ���w�� ��v��� dx �

Z�f � �w � v�� dx�

Z��rv��v� � �w � v�� dx�

�����

It is easy to see that ����� �with W � W� and v � v�� also implies the bound

sup�����

k��v��kL� ���

hence we may assume that weak convergence holds as � � �� i�e� there existsv � V� such that v� � v� This implies �� � i� j � n�

�ivj� � �iv

j weakly in L�����

vi��wj � vj�� vi�wj � vj� stronly in L�����

Note� that for the second statement we make use of the restriction n � �� Wetherefore can discuss the right�hand side of ����� in the limit � � � with theresult

Z�

f � �w � v� dx�Z�

�rv� v � �w � v� dx�

��� FUCHS et al�

The left�hand side of ����� is equal to

Z�

��v�� � ��w� dx�Z�

j��v��j�dx�

Z�

�� � j��v��j�������v�� � ���w�� ��v��� dx �

Z�

��v�� � ��w� dx�Z�

j��v��j�dx�

Z�

f�� � j��w�j���� � �� � j��v��j����g dx �

Z�

��v�� � ��w� dx�Z�

j��v��j�dx�Z�

f�� � j��w�j���� � j��v��jg dx �

Z�

��v�� � ��w� dx�Z�

�� � j��w�j����dx�Z�

�j��v��j� � j��v��j� dx�

hence

lim sup���

Z�

�W�

�E���v��� � ���w�� ��v��� dx �

lim���fZ�

��v�� � ��w� dx�Z�

�� � j��w�j����dxg�

lim inf���

Z�

�j��v��j� � j��v��j� dx �

Z�

��v� � ��w� dx�Z�

j��w�jdx�Z�

�j��v�j� � j��v�j� dx�

This shows that the weak limit v is a solution of �BVI�� Strong convergence�at least for this subsequence� is shown as follows� �BVI� together with the

Variational Methods ���

equation for v� implies

Z�

���v�� ��v��� � ���v��� ��v�� dx�

Z�

�j��v��j � j��v�j � �� � j��v��j����� ��v�� � ���v��� ��v��� dx

�Z�

f�rv�v � �rv��v�g � �v� � v� dx � ��

hence

Z�

j��v��� ��v�j�dx �

Z�

fj��v��j � j��v�j � �� � j��v��j����� ��v�� � ���v��� ��v��gdx�

Z�

f�rv�v � �rv��v�g � �v� � v� dx � � � ��

Clearly � � as � � �� for � we have

� �Z�

fj��v��j � j��v�j� �� � j��v�j���� � �� � j��v��j����g dx

�Z�

��� � j��v�j���� � j��v�j

�dx � � as � � ��

and therefore v� v as � � ���

DEFINITION ����� Suppose that n � � or � and let f � L����Rn��

a� A function v � W �� ���R

n� satisfying div v � � is said to be a weaksolution of Bingham variational inequality ����� if and only if ����� holdsfor any w from the space V��

��� FUCHS et al�

b� Let W denote the dissipative potential from Theorem ������ We say thata function v � W �

� ���Rn�� div v � �� is a weak solution of the Navier

Stokes system associated to W if and only if ����� holds for any � � V��

REMARK ����� We did not try to prove existence theorems for arbitraryDirichlet boundary data v� but we expect that the results are similar to thecorresponding results for the Navier�Stokes system�

REMARK ����� Of course we can also de�ne local solution v � W ���loc���R

n�for volume forces f � L�

loc���Rn� by requiring that ����� and ����� are valid

for functions in V� with compact support�

Next we state our regularity results concerning arbitrary weak solutions of����� and ������

THEOREM ����� Suppose that n � � or � and let f � L����Rn��Assume further that v is a weak solution of �BVI�� Then we have the followingstatements�

a� v � W ���loc���R

n��

b� If f � L��n�����Rn�� then a� holds and v � C������Rn� for any � � ��

c� In addition to the above hypothesis we assume that f � L��n��� ���Rn�for some � � � � �� Then�

i� ��v� � L�loc��� M �

ii� For n � � ��v� is a continuous function�

iii� There exists an open subset �� of � such that ��v� � � almost ever�

ywhere on ���� and ��v� � � on �

� � Moreover� v � C������ �R

n�for any � � ��

REMARK ����� All results extend to local solutions of �BVI� with volumeforces in appropriate local Morrey spaces� This is also true in the situation ofthe next Theorem�

THEOREM ����� Let the assumptions of Theorem ����� hold and let v nowdenote a weak solution of ����� with W dened in Theorem ������ Then thestatements a�� b� and c� i� are true� In place of c� ii�� iii� we obtain� thereexists an open subset �� of � such that Hn�������� � � and v � C������R

n�for any � � �� In particular� if n � �� we have �� � ��

Variational Methods ��

The proof of Theorem ����� is based on

LEMMA ����� Let n � � or � and let v denote a weak solution of �BVI� withf � L����Rn�� Then we have �rv�v � L��n����

loc ���Rn� for any � ���� �� n���

If we assume f � L��n�����Rn�� then we have �rv�v � L��n����loc ���Rn� for

any � ���� ���

Proof of Lemma ������ Since n � � or � we have by the embedding theorem

A� � kvkL�n��� ��������

Consider a ball BR�x�� � � and let v� denote the solution of the linear Stokesproblem

�������������

ZBR�x��

��v�� � ���� dx � � for all � � �W �

��BR�x���Rn�� div � � ��

v� � v��W �

��BR�x���Rn�� div v� � �

����

which satis�es the estimate

ZB��x��

jrv�j�dx � c��n�� R

�n ZBR�x��

jrv�j�dx� � � � R�

hence

ZB��x��

jrvj�dx � c��n�n� R

�nZBR�x��

jrvj�dx�Z

BR�x��jrv �rv�j�dx

o� � � � R�

����

It therefore remains to estimate the last integral on the right�hand side of�����

�� FUCHS et al�

Using ����� �w�l�o�g� we let � � ����p�k� � �� and ���� we obtain

�������������������������������������������

��

ZBR�x��

jrv �rv�j�dx �Z

BR�x��

j��v�� ��v��j�dx �

ZBR�x��

��v� � ���v�� ��v��� dx �

ZBR�x��

fj��v��j � j��v�j � f � �v� � v� � �rv�v � �v� � v�g dx �

ZBR�x��

fj��v� � v�j� jf j jv� � vj� jvj jrvj jv� � vjg dx�

��� �

Clearly

ZBR�x��

j��v��� ��v�j dx � jBR�x��j��� ZBR�x��

j��v��� ��v�j�dx���

�����

and

ZBR�x��

jf j jv� � vj dx � c��n�R�Z

BR�x��jf j�dx

����Z

BR�x��jrv �rv�j�dx

����

�����

For the remaining integral we observe

ZBR�x��

jvj jrvj jv� � vj dx �� ZBR�x��

jv� � vj�ndx���n

� ZBR�x��

�jvj jrvj� �n�n�� dx

�����n �� c��n�R

�n��

� ZBR�x��

jrv� �rvj� dx���

� ZBR�x��

jrvj�dx���� Z

BR�x��

jvj �nn��dx

�n���n �

Variational Methods ��

In case n � � we have by �����

� ZBR�x��

jvj �nn��dx

�n���n � A��

for n � � we observe

� ZBR�x��

jvj �nn��dx

�n���n � c��n�A�R

n��� �

hence

���������������

ZBR�x��

jvj jrvj jv� � vj dx � c �n�R��A�

� ZBR�x��

jrv� �rvj�dx���� Z

BR�x��

jrvj�dx���

������

Next we combine the estimates ���� � ������ to get

���������������

ZB��x��

jrvj�dx � c��n�n��

R�n �RA�

��Z

BR�x��

jrvj�dx�

Rn �R�Z

BR�x��

jf j�dxo� � � � R � dist �x�� ����

������

If f is just a function of class L�� then ������ implies the growth estimate

ZB��x��

jrvj�dx � c���� n� A�� A�� diam �� krvk�� n����

for any � � � � n��� A� � kfkL�� hence v � C������Rn�� is particular v inL�loc���R

n�� so that

�rv�v � L��n����loc ���Rn�

for � � � � n��� If f � L��n�����Rn�� then ����� implies the same result forany � � ��

��� FUCHS et al�

Proof of Theorem ������ Let us de�ne the arti�cial volume force f � f ��rv�v � L����Rn�� Then the function v is the minimizer of

J�u� �Z�

f�j��u�j� �p�k�j��u�j � f � ug dx

in the class K � fw � W �� ���R

n� � w � v on ��� div w � �g� and part a�of Theorem ����� follows from Theorem ������ b� was shown in Lemma ������Assume now that f � L��n��� ���Rn� for some � � �� Then Lemma �����implies �rv�v � L��n����

loc ���Rn� which means f � L��n��� loc ���Rn�� State�

ment c�i� follows from Theorem ����� �by noting that the proof of this theoremis the same in the presence of volume forces f as above�� c�ii� is contained inTheorem ������ c�iii� is a variant of the Main Theorem of section � �combin�ded with the remark that the Main Theorem holds for volume forces in someMorrey space��

In a similar way we can reduce the proof of Theorem ����� to the resultsobtained for �local� minimizers by establishing the following version of Lemma������

LEMMA ����� Let n � � or � and let v denote a weak solution of �����with W dened in Theorem ������ If f � L����Rn�� then we have �rv�v inL��n����loc ���Rn� for any � � � � n��� If f � L��n�����Rn�� then the same

result holds for any � � ��

Proof of Lemma ������ Let W �E� � ��jEj� � �� � jEj��p�� the Powell�Eyring

model is discussed in the same way� We de�ne v� as in ���� and get estimate

Variational Methods ���

����� ��� � has to be replaced by

���������������������������������������������������������

��

ZBR�x��

jrv �rv�j�dx �Z

BR�x��

��v� � ���v�� ��v��� dx �

ZBR�x��

�W�

�E���v�� � ���v��� ��v�� dx�

ZBR�x��

�rv�v � �v� � v� dx�

ZBR�x��

f � �v� � v� dx �

ZBR�x��

j�W�

�E���v��j j��v��� ��v�j dx�

ZBR�x��

fjf j� jrvj jvjg jv� v�j dx�

������

where W��E� � ��� jEj��p�� The last integral on the right�hand side of ������is estimated as in Lemma ������ For the �rst term we observe�����������������

ZBR�x��

j��v�� ��v��j j�W�

�E���v��jdx �

�� ZBR�x��

j��v�� ��v��j�dx���� Z

BR�x��

j�W�

�E���v��j�dx

����

������

The de�nition of W� implies �w�l�o�g� � � ��

ZBR�x��

j�W�

�E���v��j�dx � p

ZBR�x��

�� � j��v�j��p��dx �

�������������

jBR�x��j if p � �

pjBR�x��j��p� ZBR�x��

�� � j��v�j�� dx�p��

if � � p � �

������������

�njBR�x��j� jBR�x��j��p

� ZBR�x��

jrvj�dx�p��o

��� FUCHS et al�

and from ������� ������ we get

ZBR�x��

jrv� �rvj�dx � c�� ZBR�x��

jrv� �rvj�dx���n

Rn� �R

n����p��

� ZBR�x��

jrvj�dx� p��

� �R� ZBR�x��

jf j�dx���

�R��

� ZBR�x��

jrvj�dx���o

for a suitable constant c� independent of BR�x��� Next we apply Young�sinequality to obtain for any t �

���������������

ZBR�x��

jrv �rv�j�dx � c�n�R � t�

ZBR�x��

jrvj�dx

�R�Z

BR�x��

jf j�dx��� � t

p��p���Rno������

with c� independent of t and BR�x��� By combining ���� and ������ we getthe �nal estimate

���������������

ZB��x��

jrvj�dx � c�n��

R�n �R � t�

ZBR�x��

jrvj�dx�

�� � t

p��p���Rn �R�

ZBR�x��

jf j�dxo�����

being valid for any o � � R � dist �x�� ��� and t �� From ����� theclaim follows along standard lines�

Variational Methods ���

�� Some open problems and comments con�

cerning extensions

We brie�y adress some questions which might be of interest for further research�

a� nonlinear systems of Stokes type

Most of the regularity results described in the foregoing sections were based onCampanato type estimates for solutions u � W �

� ���Rn� of the linear problem�������������

div u � � on ��

Z�

��u� � ���� dx � �� � � �W �

����Rn�� div � � ��

Of course we would like to have a corresponding nonlinear variant� for � �p �� consider a function u � W �

p ���Rn� such that�������������

div u � � on ��

Z�

j��u�jp�� ��u� � ���� dx � �� � � �W �

p���Rn�� div � � ��

and prove regularity estimates in the spirit of Uhlenbeck �U�� This would im�ply optimal regularity results for power law �uid models� Moreover� we couldimprove our results on minimizers of J�u� �

R�j��u�jp � �j��u�j dx �� ���

minimizers are of class C������Rn� for any � � � � �� We conjecture that alsoTheorem ����� �local boundedness of ��u�� extends to the p�case�

b� the p�Bingham variational inequality

Starting from the dissipative potential W �E� � �pjEjp � jEj� � � p � ��

the classical Bingham variational inequality ����� has to be replaced by thefollowing more general problem� to �nd a function u with prescribed boundarydata such that div u � � and���������������

Z�

j��u�jp����u� � ���w�� ��u�� � j��w�j � j��u�j

��ru�u � �w � u� dx �Z�

f � �w � u� dx

��� FUCHS et al�

for every w satisfying div w � � and also wj�� � uj��� The time�dependentversion of p�BVI is investigated in the papers of �Ka� and �Ki�� In orderto extend Theorem ����� to weak solutions of p�BVI� we �rst have to solveproblem a�� These di�culties can be avoided by making the assumption thatp dim�� Let us further assume that the volume forces f are in the Morreyspace Lq�n�q�q

loc ���Rn� for some � � � � � where q � p�p��� and consider alocal solution v � W �

p�loc���Rn� of the p�Bingham variational inequality� In

this case v � L�loc���Rn� by Sobolev�s embedding theorem and we may apply

H&older�s inequality to see �rv�v � Lq�n�qq���np�loc ���Rn�� Hence f � f��rv�v

is in Lq�n�qqloc ���Rn�� � � minf�� �� n�pg� and v clearly is a local minimizer

of w R���pj��w�jp � j��w�j � f � w� dx subject to the constraint div w � ��

From the Main Theorem in section � we get��There is an open set �

� such that v � C������ �R

n� for any � � � and��v��x� � � for all x � �

� � on �� �� we have ��v� � � a�e�

Hence p n together with f � Lq�n�q�qloc ���Rn� is a su�cient �but unsatisfy�

ing� condition for partial C��regularity of solutions of the p�Bingham varia�tional inequality� We do not know if ��v� is still in the space L�loc��� M ��

c� analytic properties of the separating curve

As mentioned in section � it is of interest to describe the separating curve� � ����u� � o� where u is either a minimizer of the energy associated to thedissipative potential W �E� � �jEj� �p�k�jEj or just a solution of the Bing�ham variational inequality� provided � is a domain in R� �

d� evolution problems

Duvaut and Lions �DL� present an evolution model for the classical Binghamvariational inequality and prove existence theorems� the evolution problem forp�BVI is solved in the papers of Kato �Ka� and Kim �Ki�� What are the regu�larity properties of these weak solutions*

e� boundary regularity

For su�ciently regular boundary data v� �e�g� v� � C������Rn�� we expectthat Theorems ������ ����� c�i� and ����� can be improved to global bounded�ness of the strain velocity �eld� Some ideas can be found in the paper �FL��the details are carried out in �R� and �Sh�����

Variational Methods ���

f� pseudoplastic �uids

are introduced in �AM�� The dissipative potential is just W �E� � KjEj��with K � and � ���� ��� As an approximation we choose W��E� � �� �jEj������� with � � �xed but very small� Then there should hold partialC��regularity for local minimizers v� div v � �� of w R

��W����w���f �w� dx�

provided f is su�ciently regular� we refer to �R� for a further discussion�

Chapter �

Fluids of Prandtl�Eyring type

and plastic materials with

logarithmic hardening law

��� Preliminaries

According to the discussion at the beginning of the previous chapter the slow�steady state motion of a �uid of Prandtl�Eyring type in a bounded domain� � R

n � n � � or n � �� is governed by the following set of equations

div v � ������

div � � �� � � � � p������

� � ��ar sinh��j��v�j�

�j��v�j ��v������

where v � � Rn denotes the velocity �eld and ��� � are some �xed physical

parameters� We further use the symbols � and p for the Cauchy stress tensorand a suitable pressure function� respectively� Let us remark that� just fortechnical simplicity� we consider the case of zero volume forces f � � R

n �otherwise ����� has to be replaced by

div � � f� � � � � p�������

���

Variational Methods ��

As outlined in section � of chapter � the velocity �eld v is a minimizer of thevariational integral

���������J�v��� �

R�W ���v�� dx�

W �E� � ��jEjR�

�ar sinh��t� dt�

����

in classes of functions u � � Rn with div u � �� Note� that under the

assumption ������ we have to add the potentialR�f � v dx to J�v����

From this point of view it is reasonable to discuss local minimizers of thefunctional J from ���� and to analyze their regularity properties� In contrastto our previous investigations it is not immediately obvious to which naturalfunction space local minimizers should belong which means that we have tointroduce some spaces providing also existence theorems� The correct class is

�V ��� � fv � L����Rn� � div v � ��

Z�

j��v�j ln �� � j��v�j�dx ��g

and we will show �compare �FS���

THEOREM �� �� Suppose that v � �V ��� locally minimizes J����� in this

class� Then we have H�older continuity of ��v� on an open set �� � � whosecomplement is of Lebesgue measure zero� For two dimensional domains � weget �� � ��

REMARK �� �� Our investigations of Prandtl�Eyring �uids are restrictedto small values of v� i�e� we impose the restriction �rv�v � �� It is anopen problem if the results from chapter �� section �� can be extended to thePrandtl�Eyring �uid model�

In order to keep notation simple we will replace J from ���� by the functional

w Z�

G���w�� dx� G�E� � jEj ln�� � jEj��

�� FUCHS et al�

which is no serious drawback� actually our arguments work for any integrandg�j��w�j� with g�t� being C��close to t ln�� � t�� t � �� �see �FreSe�� section ��for a precise de�nition� and W �E� from ���� is of this type�

Let us now turn to the plasticity case� here � denotes the undeformed state ofan elasto�plastic body and in the case of logarithmic hardening an admissibledisplacement �eld u � � R

n is sought which minimizes

I�u��� �Z�

�G��D�u�� � �div u��� dx����

under appropriate boundary conditions �again we neglect boundary and vo�lume force terms�� The logarithmic case G�E� � jEj ln�� � jEj� lies betweenperfect plasticity and plasticity with power hardening which have been discus�sed in chapter � and �� As shown in �FreSe� the natural function space for Ifrom ���� is

X��� � fu � L����Rn� � div u � L�����Z�

j�D�u�j ln �� � j�D�u�j� dx ��g

and in the two�dimensional case local I������minimizers fromX��� are smoothin the interior of �� This result is completed by

THEOREM �� �� Consider a domain � � R� and let u � X��� denote a

local I����� minimizer� Then ��u� is of class C��� for any � � � on someopen subset �� of � such that the measure of the complement of �� is zero�

REMARK �� �� Of course it is possible to replace G��D�u�� in de�nition���� by g��j�D�u�j� with g� being in a C��sense close to t ln�� � t�� again werefer to �FreSe�� section �� for a precise statement� But since in this case thecalculations become more technical without signi�cant changes we prefer toconsider the simpli�ed model�

REMARK �� �� From the point of view of applications it might be inte�resting to consider also the case when G��D�u�� is replaced by a functiong��j�D�u�j� with g��� discontinuous at some point t� and an asymptotic beha�viour like j�D�u�j ln��� j�D�u�j�� For the necessary changes we refer to chapter�� section ��

Variational Methods ��

Chapter � is organized as follows� in section � we introduce some natural func�tion spaces for Prandtl�Eyring �uids and give a precise formulation of Theorem����� �see Theorem ������� Section � is devoted to the proof of the fact that for

a local minimizer v of ���� the functionsq� � j��v�j and � � �G

�E���v�� have

weak derivatives in L�loc���� Here we make use of some Powell�Eyring type

approximation� As a byproduct we obtain that the strain velocity is local�ly square�integrable on �� moreover� we prove a Caccioppoli�type inequality�Using a blow�up argument it is shown in section � that partial regularity holdsfor ��v� in case n � �� The two�dimensional part of Theorem ����� is discussedin section � essentially following �FreSe�� In section we give the necessarymodi�cations for plastic materials with logarithmic hardening� A �nal secti�on presents proofs of some density results for the spaces introduced in section ��

��� FUCHS et al�

��� Some function spaces related to the

Prandtl�Eyring �uid model

We de�ne the space L lnL��� in the following way� suppose that � is a boun�ded domain in Rn � Consider an open cube Q� parallel to the axes of Rn andcontaining �� For any function f � L���� we let

f �

�����f in �

� in Q� � �

and introduce the norm �compare�FreSe�� �Se���

kfkL lnL��� �ZQ�

MQ� f dx

where MQ� f is the maximal function �see �BI��

MQ� f�x� � sup f �

ZQ

j f j dx � x � Q � Q�g�

the supremum being taken over all parallel open subcubes of Q� containingthe point x� Let

L lnL��� � ff � L���� � kfkL lnL��� ��g�

The proofs of the next results concerning the space L lnL��� can be found forexample in �I�� �St�����

LEMMA ����� There exist positive constants c��n�� c��n� such that for anyf � L lnL��� the following inequality is true

c�kfkL lnL��� �Z�

jf j ln�� � �

ajf j� dx � c�kfkL lnL���������

a denoting the mean value �RQ�

j f j dx � �jQ�j

RQ�

j f j dx� jQ�j being Lebesgue�s mea�

sure�

Variational Methods ���

As an application of ����� it is easy to show that L lnL��� is a Banach space�

LEMMA ����� �De La Val�ee Poussin�Suppose that the sequence ffmg is bounded in L lnL���� Then there exists asubsequence ffmk

g and a function f � L lnL��� such that fmk� f weakly in

L�����

LEMMA ����� �see �FreSe�� Lemma ����Let Exp ��� denote the normed dual of L lnL���� For any � Exp ��� thereexists a unique measurable function g such that

�f� �Z�

gf dx� f � L lnL����

g satises

Z�

ejgjdx ��

with a suitable constant � �� Conversely� any function g with this propertygenerates an element of Exp ��� via integration�

According to the above lemma it is convenient to identify

Exp ��� � fg � � Rj g is measurable andZ�

ejgjdx �� for some � �g�

The next result is an easy calculation�

LEMMA ����� For any f � L lnL��� we have the estimates

ln ��Z�

jf j dx� j�j� �Z�

jf j ln�� � jf j� dx�����

c�kfkL lnL��� � �

ejQ�j� �

Z�

jf j ln�� � jf j� dx� ln �j�j������

By combining ����� and ����� we have

��� FUCHS et al�

LEMMA ����� Suppose that f � L����� Then f is in the space L lnL��� ifand only if

R�jf j ln�� � jf j� dx � �� Moreover� kfkL lnL��� is controled by the

logarithmic integral and vice versa�

Now we introduce some function spaces related to L lnL���� Let

V ��� � fu � L����Rn� � j��u�j � L lnL���g�

�V ��� � fu � V ��� � div u � �ii�u� � � on �g

equipped with the norm

kukV ��� � kukL���� � k��u�kL lnL����

The de�nition of the spaces Vloc�����V loc ��� should be obvious� Let us further

de�ne

�����������������������

V���� � closure of C�� ��� �Rn� in V ��� w�r�t�k � kV ����

�C �

� ���Rn� � fu � C�� ���Rn� � div u � �g�

�V � ��� � closure of

�C �

� ���Rn� in V ��� w�r�t�k � kV ����

The following result is demonstrated in the Appendix�

LEMMA ����� Suppose that � is a Lipschitz domain� Then

a� V ��� � closure of C��)��Rn� in V ��� w�r�t� k � kV ���

b��V ��� � closure of fu � C��)��Rn� � div u � �g in V ��� w�r�t� k � kV ���

c� V���� � fu � V ��� � u � � on ��g

d��V � ��� � fu � �

V ��� � u � � on ��g�

Variational Methods ���

In c�� d� of Lemma ���� u � � on �� has to be understood in the trace sense�Using the obvious inclusion

V ��� � BD���Rn������

of V ��� into the space of functions having bounded deformation �see Appendixto chapter �� we can associate to u � V ��� a trace in L�����Rn� �compare�ST��� Theorem ����� From �ST�� or �AG�� �see also �MM��� we get by quoting����� again

LEMMA ���� Let � denote a Lipschitz domain

a� For p � nn�� we have V ��� � Lp���Rn� and the inclusion is compact for

p � nn�� �

b� Let + be a subset of �� with positive �n� �� dimensional measure� thenthere is a constant c � c�n���+� such that

kukLn�n����� � cZ�

j��u�j dx

for all u � V ��� with uj� � �� In particular� this result holds for u inV�����

We introduce the energy density

G�E� � jEj ln�� � jEj�

for matrices E � M and observe

�G

�E�F � � � ln�� � jF j� � jF j

� � jF j�F

jF j � j�G

�E�F �j � � ln�� � jF j�����

���������������������

D�G�F ��E�H� � g��jF j�jF j �E � H � �E�F ��H�F �

jF j� �

�g���jF j� �E�F ��H�F �jF j�

E� F�H � M � g�t� � t ln�� � t�� t � ��

����

��� FUCHS et al�

Using ���� it is easy to see that

jEj�� � jF j � D�G�F ��E�E� � �

ln�� � jF j�jF j jEj�� jD�G�F �j � ���� �

holds for E� F � M �

According to Lemma ���� the variational integral

J�u��� �Z�

G���u�� dx

is well de�ned on the space V ��� and exactly as in �FreSe�� Theorem ���� one

can show� if �� is Lipschitz and if u� denotes a given function in�V ���� then

the variational problem

J����� min in u���V � ���

admits a unique solution� From this point of view it is reasonable to introducethe following concept�

DEFINITION ����� A function u � �V loc ��� is said to be a local minimizer

of the energy J if and only if

J�u� spt�� � J�u� �� spt��

for all � � �C �

� ���Rn��

REMARK ����� Let u � �V loc ��� denote a local minimizer� Then we have

J�u� �� � J�u� v� ��

for any open subregion � with )� � � and any v � �V � ���� In case that

�� is Lipschitz the latter inequality is equivalent to J�u� �� � J�$u� �� for all

$u � �V ��� such that u � $u on ���

Variational Methods ���

For local minimizers u we have the variational identity

Z�

� � ���� dx � �

being valid for all � � �C �

� ���Rn� where � �� �G�E

���u�� is in the space Exp ���for any open set � such that )� � �� This is a consequence of ����� Let usnow state the main result of this section already announced in Theorem ������

THEOREM ����� Suppose that n � � and let v � �V loc ��� denote a local

J minimizer�

a� If n � �� then there exists an open set �� � � such that j� � ��j � �and ��v� � C������� M � for any � � ��

b� In case n � � the result of a� holds with �� � ��

REMARK ����� For n � � the following problems seem to be of some inte�rest� �� Do singularities actually occur* If yes� optimal estimates for � � ��

should replace statement a�� �� Are local minimizers of class C������R�� forsome � � �* �� Can one prove local boundedness of ��v�* Of course� �� wouldimply ���

��� FUCHS et al�

��� Existence of higher order weak derivati�

ves and a Caccioppoli�type inequality

In this section we show for �uids of Prandtl�Eyring type that the deviator ofthe Cauchy stress tensor and also certain scalar functions involving the strainvelocity have weak derivatives in the space L�

loc����

THEOREM ����� Let u � �V loc ��� denote a local minimizer of J�v��� �R

�G���v�� dx� Then the following statements hold�

a� The deviator � � �G�E

���u�� of the Cauchy stress tensor � is in the spaceW �

��loc��� M ��

b� The functionq� � j��u�j is of class W �

��loc����

From b� and the embedding theorem it immediately follows

COROLLARY ����� Under the assumptions of Theorem ��� we have ��u� �L�loc��� M �� if n � �� and ��u� � Lp

loc��� M � for any nite p� if n � ��

Proof of Theorem ������ We �x two subregions �� �� �� �� � havingsmooth boundaries� In order to prove

q� � j��u�j � W �

� ����

we may assume that ��u� does not vanish identically on ��� According to thedensity lemma ���� b� we choose a sequence f)umg in C��)���R

n� with theproperty div )um � � and

)um � u in V ���������

satisfying in addition

k)um � ukV ���� �� k��)um�kL����� ��

Let

m � k��)um�k��L�����k)um � ukV ����

Variational Methods ��

and

Jm�v� ��� ��

� m

Z��

j��v�j� dx� J�v� ����

Moreover� we let um denote the unique minimizer of Jm��� ��� in the classfw � W �

� ����Rn� � div w � � on ��� w � )um on ���g� From ����� we deduce

���������Jm�um� ��� � Jm�)um� ��� � �

�k)um � ukV ���� � J�)um� ���

� R��G���u��dx as m��

�����

hence

supm

Jm�um� ��� ��� supm

J�um� ��� ��������

According to ����� we have

� m

Z��

j��)um�j�dx� J�)um� ��� � J�u� ����

The left�hand side is bounded from below by

� mJ�)um� ��� � J�)um� ����

hence J�)um� ��� J�u� ��� together with J�u� ��� � implies

limm�� m � �������

Next we combine ����� and estimate ����� of Lemma ����� to see

supmk��um�kL lnL���� �������

Since ��� is smooth� we have

um � )um � W �� ����R

n�� �V � ���� � V������

�� FUCHS et al�

on the other hand

supmk)umkV ���� ��

so that on account of ���� and Lemma ���� b� the sequence fumg is uniformlybounded in V ����� Using Lemma ���� a� and Lemma ����� we �nd a function

$u � �V ���� and a subsequence of fumg such that

um $u in L�����Rn�����

��um� � ��$u� weakly in L����� M ����� �

Moreover� since um � )um has zero trace on ���� we can state that

$u� u � �V � ����������

Recalling that u is a local J�minimizer the assumption u � $u together with����� would imply �by the strict convexity of J�

J�u� ��� � J�$u� ���������

But from ��� � it follows that

J�$u� ��� � lim infm�� J�um� ����

whereas ����� gives the estimate

lim supm��

J�um� ��� � J�u� ����

which is in contradiction to ������ So we have u � $u and the convergenceproperties stated in ����� ��� � hold not only for a subsequence� Moreover� wededuce from the above calculations �using ����� again�

limm��Jm�um� ��� � J�u� ��� � lim

m��J�um� ���������

limm�� m

Z��

j��um�j�dx � ��������

Variational Methods ��

After these preparations we are going to prove di�erentiability ofq� � j��u�j

following ideas of the papers �Se �� � �� ���� �FreSe� and �FS��� First of all� werecall that um is in the space W �

��loc����Rn� �a proof is given in �FS��� Theorem

���� compare also chapter �� Theorem ������� hence

�m ���G

�E���um�� � m��um� � W �

��loc���� M ��������

With this notation the Euler�Lagrange equation takes the form

Z��

�m � ��v� dx � �� v � �C �

� ����Rn��

hence there exists a pressure function �observe �m � L����� M ��

pm � L�����

satisfying the condition

Z��

pm dx � �

such that for the tensor

�m �� �m � pm �

we have

Z��

�m � ��v� dx � �� v � C�� ����R

n��������

From ������ and ������ it follows that

div �m � � a�e� on ��������

and

rpm � div �m � L�loc����R

n�������

��� FUCHS et al�

We also deduce from ������ the identityZ��

���m � ��v� dx � �� v � C�� ����R

n�� � � � � n������

Let � � C�� ����� � � � � �� and insert v � � ��um into ����� where � is

some number � n� From now on it will be convenient to drop the index m�i�e� u� �� � and p denote the quantities um� �m� �m and pm� respectively� �����then implies �we will use summationconvention� i�e� the sum is taken w�r�t� Greek or Latin indices occuring twice�

� �Z��

��� � ��� ��u� dx �

Z��

� ��� � ����u� dx�Z��

���ij ��ui �j�

dx

where we have used the fact that div u � �� This equation can be rewrittenin the form�����������������������

Z��

� ��� � ����u� dx � �Z��

���ij ��ui �j�

dx �

��Z��

���ij�i��u��j� dx�

Z��

���ij �iu� �j�

dx �

��I� � I��

���� �

and we next give estimates for I� and I�� We begin by discussing I��

I� �Z��

����ij � �ij��p��i��u��j� dx � ����� �

Z��

����ij �i��u��j� � �j�j� �i��u��i�

� dx �

Z��

����ij �i��u��j� � ���j� ��j�u����

� dx

where the last identity follows by index permutation� Hence we may write

I� �Z��

���ij��i��u��j� � �i���j�u����

� dx�

Variational Methods ���

We introduce the tensors

S��� � ���Q��� � �Q����T ��

Q���ij � �i��u��j�� �i���j�u����

and get

I� � Z��

�� ��� � S���dx�

Using

jS���j � jQ���j � c�j��u�j jr�j

for some suitable constant c� � c��n�� we �nd �recall the de�nition of � andwrite � m�

�������������������������������������������������������������������������������������

jI�j � c�� Z��

jr��u�j j��u�j��jr�j dx�

� Z��

��D�G���u�������u�� S���� dx

� c��Z��

� jr��u�j�dx����Z

��

�� jr�j� j��u�j�dx���

� �Z��

� D�G���u�������u�� ����u�� dx���

��Z��

��D�G���u���S���� S���� dx��� � ��� �

� c��Z��

� ��� � ����u� dx���

��kr�k��

Z��

� j��u�j� � j��u�j ln �� � j��u�j�� dx���

������

��� FUCHS et al�

We further have

I� �Z��

���ij �iu��j�

dx � �Z��

�ij����iu��j�

� dx

� �Z��

�ij�i��u��j�

dx

�Z��

�ij�iu����j�

dx

� �Z��

�ij�iu����j�

dx � ������

�Z��

�iju����i�j�

dx

and therefore

jI�j � c��n� ��� Z��

j��� jndx��n� Z

��

juj nn��dx

����n�

Let us now introduce the index m again� Then ���� �� ������ and the latterestimate imply

Z��

� ���m � ����um� dx �

c��n� ��h�Z��

� ���m � ����um� dx���

Jm�um� �����

��Z��

j���mjndx��n�Z

��

jumjn

n��dx����ni

Since fumg is bounded in the space V ����� the embedding lemma shows

supm

Z��

jumjn

n��dx ���

Variational Methods ���

������ guarantees boundedness of Jm�um� ���� therefore we get

Z��

� ���m � ����um� dx � c�h� �

� Z��

j���mjndx��ni

������

with c� independent of m�

We recall that �m is in the space W ���loc���� M � � L����� M � and observe n � �

to get the estimate

�Z��

j���mjndx��n � c

�Z��

jr����m�j�dx��� �

c�h�Z��

j�mj�dx���

��Z��

� jr�mj�dx���i

Standard results of �L�� or �LS� concerning the presssure function combinedwith ������ show

Z��

j�mj�dx � c�

Z��

�j�mj� � jpmj�� dx � c�

Z��

j�mj�dx�

The de�nition ������ of �m together with the bounds for �G�E

gives boundednessofR��j�mj�dx independent of m� moreover� by �����

jr�mj� � c��jr�mj� � c�� ���m � ����um��

Inserting these estimates into ������ and using Young�s inequality we �nallyobtain

supm

Z��

� ���m � ����um� dx ���������

We now specify � � � on ��� Then ������ shows

Z��

� mjr��um�j� � �

� � j��um�j jr��um�j�� dx � c��

��� FUCHS et al�

or Z��

jrq� � j��um�jj� dx � c���

After passing to a subsequence we �nd a function q � W �� ���� such thatq

� � j��um�j� q in W �� �����������

q� � j��um�j q in L������������

q� � j��um�j q a�e� on ���������

From this the desired claimq� � j��u�j � W �

� ���� will follow as soon as we

can show q �q� � j��u�j� To this purpose we write

Jm�um� ���� J�u� ��� �

�� mk��um�kL����� � J�um� ���� J�u� ��� �

�� mk��um�kL����� �

Z��

�G

�E���u�� � ���um�� ��u�� dx

�Z��

� �Z�

D�G���� t���u� � t��um��

���um�� ��u�� ��um�� ��u����� t� dt�dx

and Z��

�G

�E���u�� � ���um�� ��u��dx �

Z��

�G

�E���u�� � ���um�� ��)um�� dx�

Z��

�G

�E���u�� � ���)um�� ��u��dx � I �m � I ��m�

Variational Methods ���

From ���� we get �G�E

���u�� � Exp ���� M �� ����� implies

limm�� I ��m � ��

Clearly I �m � � which just follows from the Euler equation for u together withum � )um on ���� Using ������ and ��� � we deduceZ

��

� � j��u�j� j��um�j j��um�� ��u�j�dx �� m�

in particular �at least for a subsequence�

� � j��u�j� j��um�j j��um�� ��u�j� � a�e� on ���������

According to q � W �� ���� the function q is �nite a�e� and �see �������

j��um�j q� � � a�e� on ���

In view of ������ this implies

��um� ��u� a�e� on ���

hence q �q� � j��u�j� and the proof of Theorem ����� b� is complete�

For part a� of Theorem ����� we just recall that according to ������

supm

Z��

jr�mj�dx ��

so that there exists a tensor � � W �� ���� with the property�������

�m � � in W �� �����

�m � a�e� on ��

at least for a subsequence� But the de�nition ������ of �m together with �����and the pointwise convergence ��um� ��u� clearly gives � � ��

During the proof of partial regularity given in section � we will make essentialuse of the following result�

��� FUCHS et al�

THEOREM ����� Let u � �V loc ��� denote a local minimizer of the energy

J������ Then� for any ball BR�x�� � �� any t ���� �� and any symmetricmatrix A� we have

ZBtR�x��

jrq� � j��u�jj�dx � c��� t���R�

ZBR�x��

j��u�� Aj�dx�����

with c � c�n� denoting a nite constant�

Proof of Theorem ������ With notation introduced before we let

�� � B t���R�x��� �� � BR�x��

and recall the following facts

nq� � j��um�j

ois bounded in W �

� ���������

q� � j��um�j�

q� � j��u�j in W �

� �������� �

��um� ��u� a�e� in ���������

Since n � � or n � �� ���� � clearly implies

��um� ��u� in L����� M ��������

Let � � C������� � � � on BtR�x��� and

jr�j � c���� t�R

in ���

Then ����� gives

Z��

���m � ������$um� dx � �������

where $um�x� � um�x�� A�x� x��� �m� �m denoting a rigid motion which ischosen to satisfyZ

��

jr$umj�dx � c��n�Z��

j��$um�j�dx � c��n�Z��

j��um�� Aj�dx�������

Variational Methods ��

The existence of �m with ������ can be deduced from chapter �� Lemma ������From ������ we inferZ

��

�� ���m � ����um� dx �Z��

�� ���m � ����$um� dx �

Z��

�� ���m � ����$um� dx�Z��

�� ��pm� � ����$um� dx �

��Z��

����m � �r�� ��$um� dx�Z��

�� ��pm �����$u�m� dx �

��Z��

����m � �r�� ��$um� dx �

c��Z��

��jr�mj�dx����Z

��

jr�j� jr$umj�dx��� �

c��

��t�R

�Z��

��jr�mj�dx����Z

��

jr$umj�dx��� �

������

c��

��t�R

�Z��

��jr�mj�dx���� Z

BR�x��

j��um�� Aj�dx���

Using the de�nition of �m and equation ����� again we have as before

jr�mj� � c jr�mj� � c����m � ����um��

This together with the above estimates impliesZBtR�x��

���m � ����um� dx � c���� t���R��Z

BR�x��

j��um�� Aj�dx�

This showsZBtR�x��

jrq� � j��um�jj�dx � c���� t���R��

ZBR�x��

j��um�� Aj�dx�

On the left�hand side we may apply ���� �� on the right�hand side we use������ to get inequality ������

�� FUCHS et al�

REMARK ����� In the variational setting discussed in �FS�� we establisheda slightly stronger version of ����� which was needed to handle the case n � ��So in the framework of Prandtl�Eyring �uids we do not have to prove thisre�ned version of ������

Variational Methods ��

��� Blow�up� the proof of Theorem ����� for

n � �

We are now going to prove the �rst part of Theorem ������ So let us assume

that n � � and consider a local minimizer u � �V loc ��� of J������Recall that by

Corollary ����� we already know that ��u� is in the space L�loc��� M �� Partial

regularity is a consequence of

LEMMA ����� Fix some L � and calculate C� � C��L� as indicated inthe proof� Then� for all t ���� ��� we nd a number � � ��t� L� � such that

j���u��x��Rj � L andZ�

BR�x��

j��u�� ���u��x��Rj�dx � ��

imply

Z�

BtR�x��

j��u�� ���u��x��tRj�dx � C�t

�Z�

BR�x��

j��u�� ���u��x��Rj�dx

for any ball BR�x�� � ��

Proof of Lemma ������ Suppose that the claim is false� Then there exists asequence of balls BRk�xk� � � such that

jAkj � L� Ak � ���u��xk�Rk �Z�

BRk �xk�

j��u�� ���u��xk�Rk j�dx � ��k �

and

Z�

BtR�xk�

j��u�� ���u��xk�tRj�dx C�t

���k�

Let

vk�z� ��

�kRk�u�xk �Rkz��RkAkz � �k�z��� z � B��

��� FUCHS et al�

where �k is a rigid motion with the property

ZB�

jvkj�dz � c�

ZB�

j��vk�j�dz�

After passing to subsequences we �nd a symmetric matrix A and a functionv � W �

� �B��R��� div v � �� such that

Ak A� jAj � L�

vk v in L��B��R���

��vk� � ��v� in L��B�� M ��

�k��vk� � in L��B�� M � and a�e�

Using these facts it is easy to show that

ZB�

D�G�A����v�� �����dx � �

holds for any � � �C �

� �B��R��� hence v is of class C��B��R

�� and satis�es theCampanato�type estimate �compare chapter �� Lemma �����

Z�Bt

j��v�� ���v��tj�dz � C��L�t�Z�B�

j��v�j�dz�

We set C� � �C� and obtain the desired contradiction as soon as we can show

��vk� ��v� in L�loc�B�� M �������

Let � � C���B��� � � �� and consider functions vk �

�W �

��B��R�� with the

properties �see chapter �� Lemma ������

div vk � div �vk � ��v � vk�� � r� � �v � vk� in B������

Variational Methods ���

ZB�

jrvkj�dx � c�

ZB�

jr�j� jv � vkj�dz� c� � c��n�������

The minimizing property of u in BRk�xk� gives after scaling

ZB�

G�Ak � �k��vk�� dz �ZB�

G�Ak � �k��vk � ��v � vk�� vk�

�dz �

ZB�

G���� ���Ak � �k��vk�� � ��Ak � �k��v��

�dz�

ZB�

�k�G

�E

�Ak � �k���vk� � ���v � vk��

�� ��v � vk��r�� ��vk�� dz�

ZB�

�Z�

��k��� s�D�G��k���v � vk��r�� ��vk��

�v � vk��r�� ��vk�� ds dz

�ZB�

��� ��G�Ak � �k��vk�� dz �ZB�

�G�Ak � �k��v�� dz

�ZB�

�k�G

�E� � � dz

�ZB�

�Z�

��k��� s�D�G � � � ds dz

where we have used the fact that G is a convex function� �k just denotes thetensor

Ak � �k���vk� � ���v � vk�� � s�k��v � vk��r�� ��vk���

�� FUCHS et al�

Recalling ����� and the growth of D�G we infer

�����������������������������������������������

ZB�

�nG�Ak � �k��vk���G�Ak � �k��v��

odz �

ZB�

�k�G

�E

�Ak � �k���vk� � ���v � vk��

��

��v � vk��r�� ��vk�� dz�

ZB�

��kc�jr�j�jvk � vj�dz�

�����

The left�hand side of ����� equals

ZB�

�k��G

�E�Ak � �k��v�� � ��vk � v� dz�

ZB�

�Z�

��� s����kD�G�Ak � �k��v� � s�k��vk � v��

���vk � v�� ��vk � v�� ds dz�

hence

ZB�

�Z�

��� s����kD�G�Ak � �k��v� � s�k��vk � v��

���vk � v�� ��vk � v�� ds dz

�ZB�

�k�G

�E

�Ak � �k���vk� � ���v � vk��

��

��v � vk��r�� ��vk�� dz

Variational Methods ��

�ZB�

�k��G

�E�Ak � �k��v�� � ��vk � v� dz

�ZB�

c���kjr�j�jv � vkj�dz �

�ZB�

�kn�G�E

�Ak � �k���vk� � ���v � vk��

��

��G�E

�Ak � �k��v��o� ��v � vk��r�� dz

�ZB�

�k�G

�E�Ak � �k��v�� � ����v � vk�� dz

�ZB�

�k�G

�E

�Ak � �k���vk� � ���v � vk��

�� ��vk� dz

�ZB�

c���kjr�j�jv � vkj�dz

� �kI�k � �kI

�k � �kI

�k � c��

�k

ZB�

jr�j�jv � vkj�dz�

�� FUCHS et al�

We discuss the quantities Ijk� j � �� �� ��

jI�k j � �k���ZB�

�Z�

D�G�Ak � �k��v� � s�k��� ����vk � v��

���vk � v�� �v � vk��r��ds dzj

� ��k

ZB�

j��vk � v�jjr�j jv � vkj dz

and by the convergence properties of fvkg we get

limk��

�kI�k � ��

Next we write

��kI�k � �

�k

ZB�

n�G�E

�Ak � �k��v��� �G

�E�Ak�

o� ����v � vk�� dz

�ZB�

�Z�

D�G�Ak � s�k��v�����v�� ����v � vk��

�dz

and again

limk��

�kI�k � �

which follows from�R�D�G�Ak�s�k��v����v� �� D�G�A����v�� �� in L�

loc�B�� M �

together with ����v � vk�� � � in L��B�� M ��

Variational Methods ��

Finally we observe

��kjI�k j �

��� ��k

ZB�

n�G�E

�Ak � �k���vk� � ���v � vk��

�� �G

�E�Ak�

o�

��vk� dz���

� c�

ZB�

j��vk�j j��vk� � ���v � vk�j dz

� c��ZB�

j��vk�j�dz����Z

B�

nj��v�j� � j��vk�j�

odz���

the second factor staying bounded� whereas

ZB�

j��vk�j�dz � �� k ��

according to ������

Putting together our results we arrive at

limk��

ZB�

�Z �

���� s�D�G�Ak � �k��v� � s�k��vk � v��

���vk � v�� ��vk � v�� ds dz � ��

and the growth of D�G implies

limk��

ZB�

�j��v � vk�j�

� � jAkj� �k�j��vk�j� j��v�j� dz � ������

We are now going to establish ������ Fix some radius r ���� �� and some numberM �� ThenZ

Br

j��vk � v�j�dz �Z

Br���kj��vk�j�M �

j��vk � v�j�dz� ��k

�� FUCHS et al�

where ��k is the integral over Br � ��kj��vk�j M �� If we choose � � � on Br�then ���� obviously implies �recall ��v� � L�loc�B�� M � �

limk��

ZBr���kj��vk�j�M �

j��vk � v�j�dz � ��

For discussing the behaviour of ��k we introduce the auxiliary function

�k � ���k�q

� � jAk � �k��vk�j �q� � jAkj

��

We have j�kj � ��j��vk�j and after scaling we get from �����Z

Br

jr�kj�dz � c��� r���ZB�

j��vk�j�dz � c��� r���

so that f�kg is uniformly bounded in W �� �Br��

The de�nition of �k implies� if M is su�ciently large �independent of k�� thenwe have

�k � �

�k

q�kj��vk�j

a�e� on Br � ��kj��vk�j M �� This showsZBr���kj��vk�j�M �

j��vk�j�dz � ����k

ZBr

��k dz�k��

�����

since k�kkL��Br� stays bounded� ClearlyZBr���kj��vk�j�M �

j��v�j�dz � �� k ��

on account of �k��vk� � a�e� which together with ���� gives limk��

��k � ��

This completes the proof of �������

REMARK ����� Using Lemma ����� it is easy to show that ��u� is H&oldercontinuous near some point x� � � i�

supr��

j��u�x��rj �� andZ�

Br�x��

j��u�� ���u��x��rj�dx �� r � ��

Moreover� H&older continuity of ��u� on some open set � in � implies that u isof class C������R��� All results remain valid if we consider the case n � ��

Variational Methods �

��� The two�dimensional case

In this section we discuss Theorem ����� for the case n � � applying arguments

from �FreSe�� So let us assume that u � �V loc ��� locally minimizes J����� and

�x some smooth subdomain �� �� �� Consider a disc BR�x�� �� �� andde�ne um exactly as in section �� Using also the other notation introducedthere we recall �see ������

Z��

���m � ��v� dx � �� v � C�� ����R

��� � � �� �������

Let TR�x�� � BR�x��� BR��x��� Am �R�

TR�x��

��um� dx and

um�x� � um�x�� Am�x� x��� +m�x��

+m being a rigid motion which is chosen according to

ZTR�x��

jr umj�dx � c�

ZTR�x��

j�� um�j�dx������

We use ����� with v � ���� um� � � C���BR�x���� � � � � �� � � � in BR��x��

and jr�j � ��R�Introducing the function Qm � ���m � ����um� �from now on the sum is takenw�r�t� to Greek indices� we get just as in the proof of Theorem �����

ZBR�x��

��Qmdx � ��Z

BR�x��

����m � �r�� �� um� dx �

�� ZTR�x��

��jr�mj�dx���� Z

TR�x��

jr�j�jr umj�dx���

We also have jr�mj� � c�Qm so that by ����� and the above inequality

ZBR�x��

��Qmdx � �

Rc�� ZTR�x��

Qmdx���� Z

TR�x��

j�� um�j�dx���

������

� FUCHS et al�

Clearly �� um� � ��um��Am and r�� um� � r��um� so that application of theSobolev�Poincar�e inequality yields

� ZTR�x��

j�� um�j�dx��� � c�

ZTR�x��

jr��um�j dx������

Let hm �q� � j��um�j� Then it is easy to check that

jr��um�j � c�hmqQm

and by combining ����� and ����� we arrive at

ZBR���x��

Qm dx � c �

R�Z

TR�x��

Qmdx��� Z

TR�x��

hmqQm dx����

being valid for any disc BR�x�� �� ��� ���� is exactly the hypothesis of�FreSe�� Lemma ���� and as demonstrated there we deduce from ����� for anyq � � and any compact subdomain �� of �� there exists a constant K �K���� q� such that

ZBR�x��

Qm dx � K j lnRj�q����

is true for any disc BR�x�� �� ��� In place of ���� we may write �K � �K ����� q��Z

BR�x��

jr�mj�dx � K �j lnRj�q���� �

If we choose q � in ��� �� then the modi�cation of the Dirichlet�growthlemma given in �Fre�� p� �� � shows� �m is continuous on �� with modulus ofcontinuity independent of m�

Next we recall the formulas from section �

�m � �m � pm�� �m � m��um� ��G�E

���um�� �

� m��um� ��

j��um�j� ln �� � j��um�j� � j��um�j�j��um�j

���um��

Variational Methods �

div um � � implies tr �m � � and therefore ��pm � tr �m so that uniformcontinuity of �m gives the same for the pressure functions pm� Hence �mis continuous on �� with modulus of continuity independent of m� Using��um� ��u� a�e� we get continuity of �G

�E���u�� by Arcela�s theorem� But

�G�E

is a homeomorphism M M which shows ��u� � C����� M �� In this casethe criterion for partial regularity from Remark ����� holds at any point� Theproof of part b� of Theorem ����� is complete�

�� FUCHS et al�

�� Partial regularity for plastic materials

with logarithmic hardening

As mentioned in the beginning this type of plastic material behaviour has be�en completely investigated in the twodimensional case by Frehse and Seregin�FreSe� who also gave de�nitions of reasonable energy spaces in which soluti�ons should be located� Our main concern in this section is to prove partialregularity for three dimensions �compare Theorem ���� below� but for sake ofcompleteness we �rst recall from �FreSe� the necessary background material�

Let � denote a bounded domain in Rn �no restriction on n� and de�ne

X��� � fv � L����Rn� � div v � L����� j�D�v�j � L lnL���g

with norm

kvkX��� � kvkL���� � kdiv vkL���� � k�D�v�kL lnL����

We further let

Xloc��� � fv � � Rn jv � X��� for any open � with � � �g�

X���� � closure of C�� ���Rn� in X��� w�r�t�k � kX����

Clearly X��� is a subspace of BD���Rn�� hence for Lipschitz � there existsa trace operator X��� L�����Rn�� Quoting �FreSe� we have

LEMMA ����� Suppose � is a Lipschitz domain� Then

a� C����Rn� is dense in X����

b� u � X���� i u � X��� with u � � on ���

and from Theorem ��� of �FreSe� we also get that for smooth boundary �� anda given function u� there exists a unique minimizer of the variational integral

I�u��� �Z�

��

��div u�� �G��D�u��� dx

in the class u� �X����� We therefore de�ne

Variational Methods ��

DEFINITION ����� A function u � Xloc��� is a local extremal of the func�tional I i for all � � C�

� ���Rn�

I�u� spt �� � I�u� �� spt���

Of course Remark ����� holds with obvious modi�cations� For local minimizersu � Xloc��� it is easy to show that

Z�

hdiv u div ��

�G

�E��D�u�� � �D���

idx � �

holds for any smooth subdomain � � � with compact closure in � and anyfunction � � X����� Moreover� we have

�G

�E��D�u�� � Exp ��� M ��

THEOREM ����� Assume that n � � and let u � Xloc��� denote a localI minimizer� Then there exists an open set �� � � such that j� � ��j � �and ��u� � C������� M � for any � � �� i�e� u � C�������R

���

REMARK ����� In �FreSe� it has been shown that �� � � in case thatn � ��

Since the proof of Theorem ���� is very similar to the Prandtl�Eyring case wewill concentrate on the necessary changes�

THEOREM ����� Suppose that u � Xloc��� is a local I minimizer� Thenwe have

a� div u � W ���loc���

b�q� � j�D�u�j � W �

��loc����

If n � �� then ��u� � L�loc��� M �� For n � � we have ��u� � Lp

loc��� M � forany nite p�

�� FUCHS et al�

Proof of Theorem ����� We argue as in Theorem ����� by choosing smoothdomains �� b �� b �� The approximating sequence fumg is located in thespace C�����R

n� �see Lemma ���� a�� and satis�es

kum � ukX���� �

as well as all other requirements with V ���� replaced by X�����

Let

Im�v� ��� ��

� m

Z��

j��v�j�dx� I�v� ���

and let um denote the unique Im��� ����minimizer in the class um��W �

�����Rn��

Then we have as in section �

um u in L�����Rn������

div um � div u in L����������

��um� � ��u� in L����� M ������

mk��um�k�L����� ������

Jm�um� ��� J�u� �������

um is of class W ���loc����R

n�� hence

�m � div um���G

�E��D�um�� � m��um� � W �

��loc���� M ��

and the Euler equation for um implies �� � � � n�

Z��

���m � ��v� dx � �� v � C�� ����R

n�����

The following calculations can be found in �FreSe� but for completeness we givean outline of the arguments� We insert v � � ��um for some � � C�

� ������ � �� into ��� and use summation convention with respect to Greek and

Variational Methods ��

Latin indices repeated twice� For notational simplicity we also drop the indexm� Then ��� implies

� �Z��

��� � ��� ��u� dx �

Z��

� ��� � ����u� dx�Z��

���ij��ui �j�

dx

or equivalently

Z��

� ��� � ����u� dx � �Z��

���ij ��ui �j�

dx �

��Z��

���ij �i��u��j� dx �

Z��

���ij �iu� �j�

dx �

���I�� � �I���

In order to discuss �I�� we observe �k�k� � �� � � �� � � � � n� Then

�I�� �Z��

���ij��Di��u� � �i�

ndiv u��j�

dx �

Z��

���ij�Di��u��j�

dx �

Z��

����

Dij �

n�ij tr �����

��Di��u��j�

dx �

Z��

���Dij �

Di��u��j�

dx��

n

Z��

tr ������Di��u��i�

dx�

We have

� � �j�jk � �j�Djk �

n�jk tr ��j���

i�e�

ntr ��k�� � ��j�Djk

�� FUCHS et al�

and therefore

�I�� �Z��

h���

Dij �

Di��u��j�

� �j�Dj��

Di��u��i�

idx �

Z��

h���

Dij �

Di��u��j�

� ���Dj��

Dj��u����

idx�

the last identity trivially follows by index permutation� This may also bewritten as

�I�� �Z��

���Dij

h�Di��u��j�

� �i��D�j�u����

idx�

Let

T ��� � ���P ��� � �P ����T ��

P���ij � �Di��u��j�� �i��

D�j�u����� � � �� � � � � n�

hence

�I�� � Z��

�����D � T ���dx�

For the tensors T ��� we observe

T ��� � T ��� � P ��� � P ��� �

h�Di��u��j�� �i��

D�j�u����

ih�Di��u��j�� �i��

Drj�u��r�

i�

c�j�D�u�j�jr�j�

with a suitable positive constant c��n��

Next recall the de�nition of � � denotes the quantity m� which implies

�D ��G

�E��D�u�� � �D�u��

Variational Methods ��

hence

j�I��j � ���Z��

��D�G��D�u������D�u�� T ���� dx�

Z��

�� ���D�u� �

T ���dx��� � c�

h���Z��

��D�G��D�u������D�u�� T ���� dx

���� Z��

jr�D�u�j j�D�u�j jr�j��dxi�

Splitting �D�u� � ��u�� �ndiv u � and using Cauchy�Schwarz inequality for

the bilinear form D�G��D�u����� �� we deduce

j�I��j � c�hZ��

� � jr��u�j� �D�G��D�u������

D�u�� ���D�u��

� div ��u div ��u�dxi��

�hZ��

��� jr�j�j��u�j� �D�G��D�u���T ���� T ����

�dxi��

and get the �nal estimate for �I��

j�I��j � c��n� ��� Z��

� ���m � ����um� dx���

Jm�um� ������� �

where the index m has been introduced again� We need to estimate �I��� After

�� FUCHS et al�

integration by parts we have �dropping the index m�

�I�� � �Z��

�ij div �iu�j� dx�

Z��

�ij�iu� ���j�

dx � A� � A��

jA�j � c��Z��

� jr div uj�dx����Z

��

��jr�j�j�j�dx��� �

c��Z��

� ��� � ����u� dx����Z

��

��jr�j�j�j�dx���

jA�j ����Z��

��i�iju����j�

� �iju����i�j�

� dx��� �

���Z��

�iju����i�j�

dx��� �

c �n� ���Z��

j���jndx��n�Z

��

juj nn��dx

����n�

The remaining integrals are handled as in section � using the convergences���� � ���� the boundedness of the embedding

X���� � Lnn������Rn�

together with estimates like

�Z��

j���mjndx��n � c�

�Z��

jr����m�j�dx���

jr�mj� � c����m � ����um��

In conclusion we deduce from �� � and the latter inequalities the �nal bound

Z��

� ���m � ����um� dx � c� ������

Variational Methods �

with c� not depending on m�

Choosing � � � on �� in ���� we arrive at

supm

Z��

hjr div umj�dx � jr

q� � j�D�um�jj�

idx ��

and the proof of Theorem ���� can be completed along the lines of Theorem������ in particular� we see���������������������

q� � j�D�um�j�

q� � j�D�u�j in W �

� �����

�D�um� �D�u� a�e� in ���

div um � div u in W �� �����

����

REMARK ����� Going through the proof of Theorem ���� we see thatq� � j�D�u�j � W �

��loc��� holds up to n � �� Also Theorem ����� is valid forn � � but the four�dimensional situation seems to be of no physical interest�

The analogue of Theorem ����� is

THEOREM ����� Consider a local I minimizer u in the space Xloc����Then� for any symmetric matrix A� any ball BR�x�� and any t ���� ��� wehave Z

BtR�x��

�jr div uj� � jr

q� � j�D�u�jj�

�dx �

c�n���� t���R��Z

BR�x��j��u�� Aj�dx�

Proof of Theorem ����� Let �� � B t���R�x��� �� � BR�x�� and de�ne um as

in the proof of Theorem ����� We use ��� with���������������������

v � ����$um�

� � C�� ����� � � � on BtR�x��� jr�j � c����� t�R�

$um�x� � um�x�� A�x� x��� �m�x��

� FUCHS et al�

where �m is chosen according to ������� Then

Z��

�����m � ����um� dx �

��Z��

����m � �r�� ��$um� dx

which leads to the estimateZ��

�����m � ����um� dx �

c��R

���t�Z��

��jr�mj�dx����Z

��

j��um�� Aj�dx���

We observe

jr�mj� � c����m � ����um�

so that �using the estimates given at the end of the proof of Theorem �����

ZBtR�x��

hjr div umj� � jr

q� � j�D�um�jj�

idx �

c��R�

����t��

ZBR�x��

j��um�� Aj�dx�

On the left�hand side we use ���� together with weak lower�semicontinuityw�r�t� weak convergence in L�� for the right�hand side we observe that ����implies ��um� ��u� in L����� M � since n � ��

Now we are in the position to prove Theorem ����� consider some local I�minimizer in the space Xloc���� We want to establish the claim of Lemma����� by contradiction and use the same notation as introduced in the proof ofthis lemma� The limit function v now satis�es the equation

ZB�

�div v div��D�G�AD���D�v�� �D����

�dx � �

Variational Methods �

for all � � �W �

��B��R��� hence v is smooth� and an appropriate Campanato�

type estimate holds� Again it remains to show

��vk� ��v� in L�loc�B�� M �������

Let

G��E� ��

��tr E�� �G�ED��

For any � � C���B��� � � �� we then get �compare ������

ZB�

�nGo�Ak � �k��vk���G��Ak � �k��v��

odz �

ZB�

�k�G�

�E

�Ak � �k���vk� � ���v � vk��

�� ��v � vk��r�� dz�

ZB�

��kjr�j�jvk � vj�dz�

Using Taylor expansion for the left hand side and also bounds for D�G� we�nd that

��k��

ZB�

�hdiv �vk � v�� �

j�D�vk � v�j�� � jAD

k j� �k�j�D�vk�j� j�D�v�j�idz �

ZB�

��kjr�j�jv � vkj�dz�

ZB�

�kn�G�

�E

�Ak � �k���vk� � ���v � vk��

�� �G�

�E�Ak � �k��v��

o�

��v � vk��r�� dz�

ZB�

�k�G�

�E�Ak � �k��v�� � ����v � vk�� dz�

�� FUCHS et al�

Proceeding exactly as in the proof of Lemma ����� we get

���k � �right�hand side� �� k ��

in conclusion �after specifying ��

ZBr

j�D�v � vk�j�� � jAD

k j� �k�j�D�vk�j� j�D�v�j� dz �������

ZBr

jdiv vk � div vj�dz �� k �������

for any r � �� Clearly ����� will follow from �observe ������

ZBr

j�D�vk � v�j dz ��

Let

�k�z� ��

�k

�q� � jAD

k � �k�D�vk�j �q� � jAD

k j��

The scaled version of Theorem ���� implies

supkk�kkW �

� �Br���

for any r � � and exactly the same argument as in the proof of Lemma �����gives the desired result� Hence the blow�up lemma is also established for localI�minimizers in Xloc���� partial regularity follows�

Variational Methods ��

��� A general class of constitutive relations

Without being complete we indicate how to treat other classes of generali�zed Newtonian �uids and also elastic�plastic materials with general hardeninglaws� For simplicity we just discuss the mathematical problem of minimizingthe variational integral

J�u� �Z�

G�ru�dx

among functions u � � RM for some bounded domain � � R

n with n � �andM � �� The following assumptions are imposed on the density G � RnM R�

C��A�jEj�� �� � G�E� � C��A�jEj� � ������

G�E� � C��jEj� � ������

jEj�jD�G�E�j � C��G�E� � ������

D�G�Q��E�E� � ��� � jQj���jEj�����

A��jDG�E�j� � C��A�jEj� � ������

Here C�� C�� C�� C�� C� and � denote non�negative constants� � is some positivenumber� and �������� are required to be valid for all matrices E�Q � RnM �A � ����� ����� is a N�function with complementary function A� whichmeans that A is strictly increasing and convex satisfying in addition

limt��

A�t��t � �� limt��A�t��t ���

A��t� � k A�t� �t � t�

for suitable constants t� and k� Then the energy J is well�de�ned on theOrlicz�Sobolev space W �

A���RM � �see �A� for a de�nition�� and in �FO� we

proved using the techniques of the foregoing sections�

THEOREM ����� Let u � W �A���R

M � denote a local J minimizer�

� � FUCHS et al�

a� If n � � and � � �� then u � C������RM � for any � � ��

b� Let n � � and assume that � � ��n� Then there is an open subset �� of� such that j�� ��j � � and u � C�������R

M � for all � � ��

Let us consider some examples of N�functions A and corresponding integrandsG�

�� A��t� � tp ln�� � t�� � � p � �

In this case we may de�ne

G�E� �

���������A��jEj� if jEj � �

� E � RnM �

g�jEj� if jEj � �

where g is a quadratic polynomial which has to be chosen such that G is ofclass C�� It is easy to see that �������� hold� in particular� ���� is satis�edwith � � �� p� and we obtain

COROLLARY ����� If n � �� then the singular set is empty� In case n � �and p �� ��n we have partial regularity of local J minimizers�

�� A��t� � t ln�� � ln�� � t��� G�E� � A��jEj�Now ���� is valid for any number � �� Theorem ���� implies

COROLLARY ����� Local J minimizers are partially regular provided n �� or n � ��

�� A��t� �tR�s����ar sinh s��ds� � � � � �� G�E� � A��jEj�

Condition ���� is valid for � � �� and we get

COROLLARY ����� If � is a domain in R� � then local J minimizers u aresmooth in the interior of �� partial regularity holds in case n � � together with� � ��n�

Variational Methods � �

The third example is related to the Sutterby �uid model �see �BAH��� and for� � � it reduces to the Prandtl�Eyring �uid� in the physical setting of theSutterby model we have to discuss the variational problem �n � M � n � � or��

J�v� �Z�

j��v�jZ�

s����ar sinh s��ds dx min

subject to the constraint div v � � where v has to be taken from the space

fu � L����Rn� � j��u�j � LA���g�

THEOREM ����� Let v denote a locally minimizing velocity eld for theSutterby model� If n � �� then v is smooth� in case n � � the measure of thesingular set vanishes�

The proof of Theorem ���� essentially follows the lines of �FO�� the neces�sary adjustments are carried out in �R�� The formulation and proofs of cor�responding results for plastic materials� i�e� for minimizers of

R�fG��D�u�� �

�div u��gdx with G satisfying ��������� are left to the reader�

Appendix B

B�� Density results

For completeness we give an outline of the proof of Lemma ����� Part a� andc� can be deduced from �FreSe�� Lemma A ��� we concentrate on the proof ofd��

For r � � let

DrL lnL��� � fv � L����Rn� � div v � Lr���� j��v�j � L lnL��g�

�D r

L lnL��� � closure of C�� ���Rn� with respect to the norm

kvk � kvkL���� � k div vkLr��� � k��v�kL lnL����

Then it was shown in �FreSe� �recall � is a Lipschitz domain� assuming r � nn�� �

DrL lnL��� � closure of C����Rn� w�r�t� the norm k � k in Dr

L lnL��������

�D r

L lnL��� � fv � DrL lnL��� � v � � on ��g������

Let us abbreviate

W ��V � ���� W � fv � �

V ��� � v � � on ��g�

We assume that there is a velocity �eld u� � W � u� �� W � In order to get acontradiction we select a linear functional � W � such that

�u�� � �� �v� � � for v � W�����

� �

Variational Methods � �

and introduce the operator

A � W �� ��� � f� � L���� M � � tr � � �� j�j � L lnL���g�

A�v� � ��v��

the norm on�� ��� being de�ned through

k�k ����

� k�kL lnL����

With suitable positive constants c�� c� we have

c�kvkV ��� � kA�v�k ����

� c�kvkV ���

and therefore

g � A� W � R� g��� � �A�����

is a continuous linear functional on the subspace A� W � of�� ��� satisfying

jg���j � kk kA���kV ��� � �

c�kk k�k

�����

Let G � �� ���� denote some extension of g with the property

kGk � kgk � �

c�k�k�

Recalling the de�nition of Exp ��� and also the statement of Lemma ����� wehave the representation

G��� �Z�

� � � dx� � � �� ����

with a unique tensor � � L���� M �� tr � � �� � � Exp ��� M ��

� � FUCHS et al�

From the construction we then deduce

�v� � g�A�v�� � G�A�v�� �Z�

� � ��v� dx

for any v � W � The functional vanishes on the space W � quoting �L�� or �LS�we �nd a pressure function p � L�����

R�p dx � �� such that

�����Z�

p div v dx �Z�

� � ��v� dx

for any v � �W �

����Rn�� � � Exp ��� M � implies � � Lq��� M � for any �nite q

so that

p � Lq���� q ��������

Taking q n we get from �����

Z�

p div v dx �Z�

� � ��v� dx� v � �D

rL lnL��������

where r � qq�� � n

n�� � Clearly u� � DrL lnL��� and u� � � on �� �recall

div u� � ��� by ����� a sequence um � C�� ���Rn� exists such that

um u� in DrL lnL����

���� implies

Z�

p div um dx �Z�

� � ��um� dx�

Due to div u� � � we must have kdiv umkLr��� �� in conclusion

Z�

� � ��u�� dx � ��

hence �u�� � � which is the desired contradiction��

Bibliography

�A� Adams� R�A�� Sobolev spaces� Academic Press� New York �� �

�An� Anzellotti� G�� On the existence of the rates of stress and displacementfor Prandtl�Reuss plasticity� Quarterly J� of Appl�Math� �� ���������������

�AG�� Anzellotti� G�� Giaquinta� M�� Existence of the displacement �eld foran elastic�plastic body subject to Hencky�s law and von Mises yieldcondition� Manus�Math� �� ������� �������

�AG�� Anzellotti� G�� Giaquinta� M�� On the existence of the �elds of stres�ses and displacements for an elastic�perfectly plastic body in staticequilibrium� J�Math� Pure Appl� � ������� ��������

�AG�� Anzellotti� G�� Giaquinta� M�� Convex functionals and partial regula�rity� Arch�Rational Mech�Anal� ��� ��� ������� ����� ��

�AM� Astarita� G�� Marrucci� G�� Principles of non�Newtonian �uid mecha�nics� McGraw�Hill� London� �� ��

�BAH� Bird� R�� Armstrong� R�� Hassager� O�� Dynamics of polymeric liquids�Volume � Fluid mechanics� John Wiley� Second Edition ��� �

�BF� Bensoussan� A�� Frehse� J�� Asymptotic Behaviour of Norton�Ho��sLaw in Plasticity Theory and H��regularity� Res�Notes Appl�Math��� ������� ����

�BI� Bojarski� B�� Iwaniec� T�� Analytical foundations of the theory ofquasiconformal mappings in Rn � Ann�Acad�Sci�Fenn�Ser�A�I� � �������� �����

�BS� Br�ezis� H�� Stampacchia� G�� Sur la r�egularit�e de la solutiond�inequations elliptiques� Bull�Soc�Math�France � ������ �������

� FUCHS et al�

�CR� Ca�arelli� L�A�� Riviere� N�M�� On the Lipschitz character of the stresstensor when twisting an elastic�plastic bar� Arch�Rational Mech�Anal�� ��� ��� �����

�C� Campanato� S�� Proprieta di H&olderianita di alcune classi di funzioni�Ann� Scuola Norm�Sup� Pisa � ������ � �����

�CLT� Carriero� M�� Leaci� A�� Tomarelli� F�� Strong solutions for an elasticplastic plate� Calc� of Var� � ������� ��������

�D� Demengel� F�� Compactness theorems for spaces of functions withbounded derivatives and applications to limit analysis problems inplasticity� Arch�Rational Mech�Anal� �� ������� �������

�DT� Demengel� F�� Temam� R�� Convex functions of a measure and appli�cations� Indiana Univ�Math�J� �� ������� �� ���

�DL� Duvaut� G�� Lions� J�L�� Inequalities in Mechanics and Physics� Sprin�ger Grundlehren ���� Springer Verlag� Berlin� �� �

�DS� Dunford� N�� Schwartz� J�T�� Linear Operators� Vol��� General Theory�Wiley� New York� �� �

�ET� Ekeland� I�� Temam� R�� Convex Analysis and Variational Problems�North�Holland� Amsterdam� �� �

�EG� Evans� L�C�� Gariepy� R�F�� Blow�up� compactness and partial regula�rity in the calculus of variations� Ind� Univ� Math� J� � No��� ���� ������ ��

�EK� Evans� L�C�� Knerr� B�� Elastic�plastic plane stress problems� Ap�pl�Math�Optim� ��� ��� ��������

�E� Eyring� H�J�� Viscosity� plasticity� and di�usion as examples of abso�lute reaction rates� J� Chemical Physics � ������ ��������

�Fi� Fichera� G�� Existence theorems in elasticity� and unilateral cons�traints in elasticity� Handbuch der Physik VI a� �� ����� Springer�Verlag� Berlin� �� ��

�Fre� Frehse� J�� Twodimensional variational problems with thin obstacles�Math�Z� ��� ��� �� � ������

Variational Methods �

�FreSe� Frehse� J�� Seregin� G�� Regularity for solutions of variational problemsin the deformation theory of plasticity with logarithmic hardening�Preprint No����� SFB �� Universit&at Bonn ������ to appear inProc� St�Petersburg Math�Soc� ���� ��

�F� Friedman� A�� Variational principles and free boundary problems� NewYork� Wiley�Interscience� �� ��

�Fri� Friedrichs� K�O�� On the boundary value problems of the theory ofelasticity and Korn�s inequality� Ann� Math� �� ���� �� ����� ��

�Fu�� Fuchs� M�� Regularity for a class of variational integrals motivated bynonlinear elasticity� Asymp� Analysis � ������� ������

�Fu�� Fuchs� M�� On stationary incompressible Norton �uids and some ex�tensions of Korn�s inequality� J� Analysis Appl� Z�A�A� �� ������� ������ �

�Fu�� Fuchs� M�� On quasi�static non�Newtonian �uids with power law�Math� Meth� Appl� Sciences �� ������ ���������

�Fu�� Fuchs� M�� Quasi�static non�Newtonian �uids� Vorlesungsreihe No���� SFB �� Universit&at Bonn�

�Fu� Fuchs� M�� On a class of variational problems related to plasticity withpolynomial hardening� Applicable Analysis � ������ ���� ��

�Fu� Fuchs� M�� Variational methods for quasi�static non�Newtonian �uids�Zap� Nauchn� Sem� St�Petersburg Odtel Mat� Inst� Steklov �POMI���� ������ ���

�FGR� Fuchs� M�� Grotowski� J�F�� Reuling� J�� On variational models forquasistatic Bingham �uids� Math� Meth� Appl� Sciences �� ��������������

�FL� Fuchs� M�� Li� G�� Global gradient bounds for relaxed variational pro�blems� Manus� Math� �� ���� �� �� �����

�FO� Fuchs� M�� Osmolovski� V�� Variational integrals on Orlicz�Sobolevspaces� J� Anal� Appl� Z�A�A� � �� ������� �������

�FR� Fuchs� M�� Reuling� J�� Partial regularity for certain classes of poly�convex functionals related to non�linear elasticity� Manus� Math� � ������ �����

� � FUCHS et al�

�FS�� Fuchs� M�� Seregin� G�� Some remarks on non�Newtonian �uids in�cluding nonconvex perturbations of the Bingham and Powell�Eyringmodel for viscoplastic �uids� M�AS ���� �� �������

�FS�� Fuchs� M�� Seregin� G�� Regularity results for the quasi�static Bing�ham variational inequality in dimensions two and three� Math�Z� �� ������� �����

�FS�� Fuchs� M�� Seregin� G�� A regularity theory for variational integralswith L lnL�growth� Calc� of Var� ������ �� � ���� �

�FS�� Fuchs� M�� Seregin� G�� Variational methods for �uids of Prandtl�Eyring type and plastic materials with logarithmic hardening�Math�Meth�Appl�Sciences �� ������� �� ����

�Ga�� Galdi� G�� An introduction to the mathematical theory of the Navier�Stokes equations� Vol� I� Springer Tracts in Natural PhilosophyVol� ��� Springer Verlag� New York� �����

�Ga�� Galdi� G�� An introduction to the mathematical theory of the Navier�Stokes equations� Vol� II� Springer Tracts in Natural PhilosophyVol�� ��� Springer Verlag� New York� �����

�Gi� Giaquinta� M�� Multiple integrals in the calculus of variations andnonlinear elliptic systems� Ann�Math�Studies� No���� Princeton Uni�versity Press� Princeton � N�Y�� �����

�GM� Giaquinta� M�� Modica� G�� Nonlinear systems of the type of the sta�tionary Navier�Stokes system� J�Reine Angew�Math���� ������� � ������

�HK� Hardt� R�� Kinderlehrer� D�� Elastic�Plastic deformation� Ap�pl�Math�Optim� �� ������� �������

�IS� Ionescu� I�R�� Sofonea� M�� Functional and numerical methods in vis�coplasticity� Oxford University Press� Oxford� �����

�I� Iwaniec� T�� Lp�theory of quasiregular mappings� Quasiconformalspace mappings� A collection of surveys ��������� Springer LectureNotes in Math� Vol� ���� Springer Verlag �����

�Ka� Kato� Y�� Variational inequalities of the Bingham type in three di�mensions� Nagoya Math�J� ��� ������� ����

Variational Methods � �

�Ki� Kim� J�U�� On the initial�boundary value problem for a Bingham �uidin a three�dimensional domain� Trans� A�M�S� ��� ���� �� �� ��

�Kl� Klyushnikov� V�D�� The mathematical theory of plasticity�Izd�Moskov�Gos�Univ�� Moscov� �� ��

�KT� Kohn� R�� Temam� R�� Dual spaces of stresses and strains with appli�cations to Hencky plasticity� Appl�Math�Optim� �� ������� ����

�Ko�� Korn� A�� Die Eigenschwingungen eines elastischen K&orpers mit ru�hender Ober�&ache� Akad�Wiss� M&unchen� Math��Phys� Kl�� Ber� ������� �������

�Ko�� Korn� A�� &Uber einige Ungleichungen� welche in der Theorie der ela�stischen und elektrischen Schwingungen eine Rolle spielen� Bull� ist�Cracovie Akad� umiejet� Classe sci�math�nat� ������� �� ���

�L�� Ladyzhenskaya� O�A�� The mathematical theory of viscous incompres�sible �ow� Gordon and Breach� ����

�L�� Ladyzhenskaya� O�A�� On nonlinear problems of continuum mecha�nics� Proc�Internat�Congr�Math� �Moscow ���� �Nauka�� Moscowl��� �� �� English transl� in Amer�Math�Soc�Transl� ��� � ������

�L�� Ladyzhenskaya� O�A�� New equations for the description of motion ofviscous incompressible �uids and global solvability of boundary va�lue problems for them� Trudy Mat�Inst�Steklov ��� ��� �� ������English transl�Proc�Steklov Inst�Math� ��� ��� ��

�L�� Ladyzhenskaya� O�A�� On some modi�cations of the Navier�Stokesequations for large gradients of velocity� Zap�Nauchn�Sem� LeningradOdtel�Mat�Inst�Steklov �LOMI� ������ ������ English transl� inSem�Math�Inst�Leningrad ������

�LS� Ladyzhenskaya� O�A�� Solonnikov� V�A�� Some problems of vectoranalysis� and generalized formulations of boundary value problemsfor the Navier�Stokes equations� Zap�Nauchn�Sem� Leningrad Odtel�Mat�Inst�Steklov �LOMI� �� ����� ��� �� Engl�transl� in J�SovietMath� �� No�� ��� ���

�LU� Ladyzhenskaya� O�A�� Ural�tseva� N�N� Linear and Quasilinear Ellip�tic Equations� Nauka� Moskow� ���� English translation� AcademicPress� New York ���� Second Russian edition� Nauka� Moscow� �� ��

��� FUCHS et al�

�MSC� Mathies� H�� Strang� G�� Christiansen� E�� The saddle point of a di�e�rential program� Energy Methods in Finite Element Analysis� volumededicated to Professor Fraejs de Veubeke % Eds� R� Glowinski� E� Ro�din� O�C� Zienkiewicz� New York� Wiley� �� ��

�MM�� Mosolov� P�P�� Mjasnikov� V�P�� On well�posedness of boundary valueproblems in the mechanics of continuous media� Mat� Sbornik �� �������� ��� ������ Engl�translation in Math� USSR Sbornik � � no� ���� ��� � ����

�MM�� Mosolov� P�P�� Mjasnikov�V�P�� Mechanics of rigid plastic media���Nauka�� Moscow� ���� �Russian��

�MNRR� M�alek� J�� Nec�as� J�� Rokyta� M�� R�u�zi�cka� M�� Weak and Measure�

valued Solutions to Evolution Partial Di�erential Equations� AppliedMathematic and Mathematical Computation vol� ��� Chapman andHall� ����

�NH� Nec�as� J�� Hlav�a�cek� I�� Mathematical theory of elastic and elasto�plastic bodies� an introduction� Elsevier Publishing Company�Amsterdam�Oxford�New York� �����

�N� Norton� M�� The creep of steel at high temperature� McGraw Hill�New York� �����

�P� Pileskas� K�I�� On spaces of solenoidal vectors� Zap�Nauch�Sem� Le�ningrad Otdel�Mat�Inst�Steklov �LOMI� � ������� �� ����� Englishtrans� in J�Soviet Math� �� ������� no��

�Pr� Prager� W�� Introduction to the mechanics of continua� Dover Publi�cations� New York �� ��

�PE� Powell� R�E�� Eyring� H�� Mechanism for relaxation theory of viscosity�Nature �� ������� �� �����

�RS� Repin� S�I�� Seregin� G�A�� Existence of a weak solution of theminimax problem arising in Coulomb�Mohr plasticity� Amer�Math�Soc�Transl���� Vol��� ������ ��������

�R� Reuling� J�� thesis Saarbr&ucken ��� �

�Se�� Seregin� G�A�� Variational�di�erence scheme for problems in the me�chanics of ideally elastoplastic media� Zh�Vychisl�Mat� i Mat�Fiz� ������� �� ��� �in Russian�� English translation� U�S�S�R� Com�put�Math� and Math�Phys� � ������ �����

Variational Methods ���

�Se�� Seregin� G�A�� Di�erential properties of weak solutions of nonlinearelliptic systems arising in plasticity theory� Mat�Sb� �N�S�� ��� ���� � �� � ������ ������� �in Russian�� English translation� Math� USSR�Sb� � ���� �� ��������

�Se�� Seregin� G�A�� Di�erentiability of local extremals of variational pro�blems in the mechanics of perfect elastoplastic media� Di�erentsial�nyeUravneniya �� ���� ���� �� ���������� English transl� Di�erentialEquations �� ���� �� ���������

�Se�� Seregin� G�A�� On di�erential properties of extremals of variatio�nal problems arising in plasticity theory� Di�erentsial�nye Uravneniya� ������� ��������� �in Russian�� English translation� Di�erentialEquations � �������

�Se� Seregin� G�A�� On regularity of weak solutions of variational problemsin plasticity theory� Dokl�Acad�Sci� ��� ������� ��������� �in Russi�an�� English translation� Soviet Math�Dokl� �� �������

�Se� Seregin� G�A�� On the regularity of weak solutions of variational pro�blems in plasticity theory� Algebra i Analiz � ������� ������� �in Rus�sian�� English translation� Leningrad Math� J� � �������

�Se � Seregin� G�A�� On regularity of minimizers of certain variational pro�blems in plasticity theory� Algebra i Analiz � ������� ������� �in Rus�sian�� English translation� St�Petersburg Math�J� � ������� ���������

�Se�� Seregin� G�A�� On di�erentiability properties of the stress tensor inCoulomb�Mohr plasticity� Algebra i Analiz � ������� ������ �in Rus�sian�� English translation� St� Petersburg Math� J� � ������� �� ��� ��

�Se�� Seregin� G�A�� Di�erentiability properties of weak solutions of certainvariational problems in the theory of perfect elasticplastic plates� Ap�pl�Math�Optim� �� ������� �� ����

�Se��� Seregin� G�A�� Twodimensional variational problems in plasticity theo�ry� Izv�Russian Academy of Sciences � ������ � ���� �in Russian��English translation in Izvestiya� Mathematics �� no� � ������ � �����

�Se��� Seregin� G�A�� On the di�erentiability of local extremals of va�riational problems in the mechanics of rigidly viscoplastic media�Izv�Vyssh�Uchebn�Zaved Mat�No� �� ���� ���� �� ����� �in Russi�an�� Engl�translation� Sov�Math� �Iz�VUZ� �� ���� ��

��� FUCHS et al�

�Se��� Seregin� G�A�� On di�erential properties of extremals of variatio�nal problems of the mechanics of viscoplastic media� Proc� SteklovInst� Math� � ������� �� �� �

�Se��� Seregin� G�A�� Continuity for the strain velocity tensor in two�dimensional variational problems from the theory of the Bingham�uid� Preprint No����� SFB �� Universit&at Bonn�

�Se��� Seregin� G�A�� A local estimate of maximum of the module ofthe deviator of strain tensor in elastic body with linear hardening�Zap�Nauchn�Sem� St�Petersburg Otdel� Mat�Inst� Steklov �POMI� ���������� � �� �

�Se�� Seregin� G�A�� Some remarks on variational problems for functionalswith L lnL�growth� Zapiski Nauchn�Sem� POMI� Petersburg Odtel�Steklov Math�Inst� ��� ������� ���� ��

�Ser� Serrin� J�� Mathematic principles of classical �uid mechanics� In theEncyclopedia of Physics� Vol� VIII%�� edited by S� Fl&ugge� SpringerVerlag� Berlin ����

�Sh�� Shilkin� T�N�� Regularity up to the boundary for solutions to someboundary�value problems of the generalized Newtonian �uid theory�Problemy Mat�Analiza � ���� �� ����� �in Russian��

�Sh�� Shilkin� T�N�� On problems of the generalized Newtonian �uidstheory with dissipative potential of subquadratic growth� ProblemyMat�Analiza � ������ �in Russian��

�So�� Sobolev� S�L�� Introduction to the Theory of Cubic Formulas� Nauka�Moscow� �� � �in Russian��

�So�� Sobolev� S�L�� Some Applications of Functional Analysis to Mathema�tical Physics� Nauka� Moscow� ���� �in Russian��

�St�� Stein� E�M�� Note on the class L logL� Studia Math� �� ������ �������

�St�� Stein� E�� Singular integrals and di�erentiability properties of functi�ons� Princeton U�P�� Princeton� �� ��

�Str� Strauss� M�J�� Variations of Korn�s and Sobolev�s inequality� Berkeleysymp� on P�D�E�� AMS Symposia �� ��� ��� �� �����

Variational Methods ���

�ST�� Strang� G�� Temam� R�� Duality and relaxations in the theory of pla�sticity� J�M�echanique �� ������� ����

�ST�� Strang� G�� Temam� R�� Functions of bounded deformation�Arch�Rational Mech�Anal� ������� ����

�Su� Suquet� P�� Existence et r�egularit�e des solutions des equations de laplasticit�e parfaite� Th,ese de �e Cycle� Universit�e de Paris�VI� �� ��Also C�R�Acad�Sci�Paris� Ser�D� �� ��� ��� ����������

�T� Temam� R�� Probl,emes math�ematiques en plasticit�e� Paris� Gauthier�Villars� ����

�U� Uhlenbeck� K�� Regularity for a class of nonlinear elliptic systems�Acta Math� ��� ��� �� ��������

�Z� Zeidler� E�� Nonlinear functional analysis and its applications� Vol� IV�Springer Verlag� Berlin� ��� �

authors� addresses

Martin FuchsUniversit&at des SaarlandesFachbereich � MathematikPostfach � �� �D���� Saarbr&uckenGermanye�mail� fuchs�math�uni�sb�de

Gregory SereginV�A� Steklov Mathematical InstituteSt� Petersburg branchFontanka � St� Petersburg� ������Russiae�mail� seregin�pdmi�ras�ru