Elliptic Systems, Singular Sets and Dini Continuity

Frank Duzaar,1,* Andreas Gastel,1 and Giuseppe Mingione2

1Mathematisches Institut der Friedrich-Alexander-Universitat,Erlangen-Nurnberg, Erlangen, Germany

2Dipartimento di Matematica, Universita di Parma,Parma, Italy

ABSTRACT

We estimate the size of the singular set of solutions to non-linear elliptic systems

of the form

�div aðx; u;DuÞ ¼ bðx; u;DuÞ

where the vector field a satisfies a Dini-type continuity condition with respect tothe variables ðx; uÞ.

Key Words: Elliptic systems; Singular set; Hausdorff dimension; Dini

continuity.

Mathematics Subject Classification: 35J60; 35D10.

*Correspondence: Frank Duzaar, Mathematisches Institut der Friedrich-Alexander-

Universitat, Erlangen-Nurnberg, Bismarckstr. 11=2, 91054 Erlangen, Germany; Fax: +49-9131-8522081; E-mail: duzaar@mi.uni-erlangen.de.

COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS

Vol. 29, Nos. 7 & 8, pp. 1215–1240, 2004

1215

DOI: 10.1081/PDE-200033734 0360-5302 (Print); 1532-4133 (Online)

Copyright # 2004 by Marcel Dekker, Inc. www.dekker.com

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1. INTRODUCTION

We consider an elliptic system in divergence form of the following simple type:

�div aðx;DuÞ ¼ 0 in O: ð1:1Þ

Here O � Rn is a bounded domain, n � 2, u mapping O into RN . The continuousvector field a : O�RN �MN�n ! MN�n, is assumed to satisfy the followinggrowth, ellipticity and continuity assumptions (see Sec. 2):

jDzaðx; zÞj � L ; njxj2 � Dzaðx; zÞx� x ; ð1:2Þ

jaðx; zÞ � aðx0; zÞj � Ljx� x0jað1þ jzjÞ ; ð1:3Þ

where x; x0 2 O, z, x 2 MN�n and a 2 ð0; 1�. Here MN�n denotes the space of allN � n matrices.

Without supposing any additional structure on the vector field a, it is knownthat one cannot expect everywhere regularity of weak solutions u 2 W1;2ðO;RN Þ;indeed, one can only prove partial regularity, that is Du is regular on an open subsetO0 � O of full measure,

u 2 C1;alocðO0;RN Þ and jOnO0j ¼ 0: ð1:4Þ

It is remarkable that the degree of Holder continuity of the gradient of the solution u

is the same exhibited by the vector field a in (1.3). This result – which can be consid-ered as an analogue of the well-known Schauder estimates for linear elliptic systemsin divergence form – is presented in Duzaar and Grotowski (2000). A similar sensi-tivity with respect to the regularity of the coefficients a is then shown by the size ofthe singular set OnO0. Denoting by dimHðAÞ the Hausdorff dimension of a setA � Rn, it turns out that

dimHðOnO0Þ < n� 2a: ð1:5ÞThis estimate is classical when differentiability of a holds with respect to x (seeCampanato, 1983), while the case a 2 ð0; 1Þ has been considered only more recentlyin Mingione (2003a,b).

The aim of this paper is to establish a similar estimate for the Hausdorff dimen-sion of the singular setOnO0 in a borderline case. The situation is as follows: For bothelliptic equations (Burch, 1978; Kovats, 1997) and non-linear elliptic systems in diver-gence form Duzaar and Gastel (2002) it is still possible – without assuming Holdercontinuity of the coefficients a – to prove continuity of Du (partial regularity in thecase of systems) under the weaker assumption of Dini continuity of a. Here Dini con-tinuity means that there exists a modulus of continuity m : ½0;1Þ ! ½0;1Þ such that

jaðx; zÞ � aðx0; zÞj � Lmðjx� x0jÞð1þ jzjÞ ;

MðrÞ :¼Z r

0

mðrÞr

< 1 for some r > 0 ð1:6Þ

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holds for any x 2 O and z 2 MN�n. Moreover, in this case, the gradient Du hasmodulus of continuity r 7!MðrÞ in a neighborhood of any regular point x 2 O0.Observe that in the case mðrÞ ¼ ra this result reproduces the result stated in (1.4).

Our present work, partially relying on the method of Mingione (2003a), origi-nates in the search for an analogue of the estimate (1.5) when the Holder continuityassumption (1.3) is replaced by a suitable Dini continuity hypothesis. Since we arenot dealing with a modulus of continuity exhibiting the decay ra for somea 2 ð0; 1� we cannot expect to obtain a dimension reduction of the singular set asin (1.5). Nevertheless, it turns out that an analogue of (1.5) still holds. In order totreat such a borderline case we have to use Hausdorff measures which are definedin terms of infinitesimal functions rather than positive real numbers. We refer tothe original paper of Hausdorff (1918). The main results of the present paper arecontained in Theorem 2.1 and Theorem 2.2 (see Sec. 2). Again, our results give backthe estimate in (1.5) when specialized to the case mðrÞ � ra. The role of Dinicontinuity as the limit case of Holder continuity, and its applications to PDE andPotential Theory, has been recently emphasized in Taylor (2000) and it appears indifferent settings: fully non-linear equations (Kovats, 1997), Neumann typeproblems (Lieberman, 2002) and Potential Theory on Riemannian manifolds(Mitrea and Taylor, 2003), to name a few. In particular, in this last paper, Dini-typecontinuity assumptions similar to ours are considered.

Throughout the paper, we shall consider systems of the more general type

�div aðx; u;DuÞ ¼ bðx; u;DuÞ: ð1:7Þ

2. ASSUMPTIONS, STATEMENTS AND PRELIMINARIES

Notation. In the following O � Rn denotes a bounded domain. We writeBR ¼ BRðx0Þ :¼ fx 2 Rn : jx� x0j < Rg. Moreover, an stands for the volume ofthe unit ball in Rn. When not specified otherwise, all balls considered will be con-centric. By c we denote a positive, finite constant, not necessarily the same in anytwo occurrences; the relevant dependencies of c will be specified. If BR YO and v

is a locally integrable function on O, we denote the mean value of v on BR by

ðvÞR � ðvÞx0;R :¼ �Z

BR

v dx � 1

anRn

ZBR

v dx ;

Finally, for a (vector valued) function v 2 LqðOÞ, q 2 ½1;1�, we write kvkLqðOÞ ¼ kvkq.

Assumptions on the System. We shall deal with systems as in (1.7) under thefollowing assumptions concerning the coefficients a and the right-hand side b:

jDzaðx; u; zÞj � L;

njxj2 � Dzaðx; u; zÞx� x;

jaðx; u; zÞ � aðx0; u0; zÞj � Lmðjx� x0j þ ju� u0jÞð1þ jzjÞ ;

:ð2:1Þ

8>>>><>>>>:

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and

jbðx; u; zÞj � Cð1þ jzj2Þ ; ð2:2Þ

for any z; x 2 MN�n; u; u0 2 RN and x; x0 2 O. Here, L 2 ½1;1Þ; n 2 ð0;L�; n � 2;

N � 2, and a : O�RN �MN�n ! MN�n and b : O�RN �MN�n ! RN arecontinuous functions.

Assumptions on l. With regard to the modulus of continuitym : ½0;1Þ ! ½0;1Þ we assume that it is an increasing, concave (which can alwaysbe supposed for a modulus of continuity) function satisfying mð0Þ ¼ 0 andmð1Þ ¼ 1. We also have to require that ð0;1Þ 3 r 7! r�gmðrÞ is non-increasing forsome 0 < g < 1 (that is, m is g-decreasing); this assumption is mainly required tobe under validity of the regularity results of Duzaar and Gastel (2002). Mostimportantly, we require

MðrÞ ¼Z r

0

mðrÞr

dr < 1:

A function m satisfying the previous inequality is said to satisfy the Dini condition.If m is the modulus of continuity of a function f : O ! MN�n, i.e.,jfðxÞ � fðyÞj � Lmðjx� yjÞ for every x; y 2 O, then f is called Dini continuous. Sincewe are not interested in treating the case of a Holder modulus of continuity, whichhas already been treated in Mingione (2003a,b), we assume that for any a 2 ð0; 1Þthere exists A ¼ Aða; mðÞÞ 2 ð1;1Þ such that

lim supr!0

ra

mðrÞ � A:

It is easily checked that the previous assumption implies, for another constant stilldenoted by A, the following:

ra � AmðrÞ 8r 2 ½0; 1�: ð2:3Þ

Note that this condition is technical and really non-restrictive, since m represents amodulus of continuity. Moreover, it is automatically satisfied for a ¼ 1 with A ¼ 1because m is concave, i.e., we have

r � mðrÞ 8r 2 ½0; 1�: ð2:4Þ

Smallness Condition. We are dealing with systems of type (1.7) with a righthand side of critical growth as formulated in (2.2). Therefore, we shall always dealwith bounded weak solutions u satisfying the smallness assumption (with C from(2.2))

2Cjjujj1 < n: ð2:5Þ

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Such a smallness assumption is classical (and necessary) when dealing with suchsystems (see comments in Mingione, 2003b). For future purposes we define

d :¼ n � 2Cjjujj1 > 0: ð2:6Þ

Hausdorff Measures. Let o : ½0;1Þ ! ½0;1Þ be an increasing, continuousfunction with oð0Þ ¼ 0. Then, the generalized spherical Hausdorff measure onRn associated to o is defined by

HoðAÞ ¼ limd&0

inf

(X1i¼1

oðriÞ : A �[1i¼1

BriðyiÞ; ri � d for i 2 N

)

for any subset A � RN (see Hausdorff, 1918, Definition 2). Observe that whenoðrÞ ¼ rs, s > 0, this is nothing but the usual spherical Hausdorff measure Hs.The Hausdorff measure lo associated to o is defined as

loðAÞ ¼ limd&0

inf

(X1i¼1

oðdiam AiÞ : A �[1i¼1

Ai; Ai � Rn

diam Ai � d for i 2 N

):

Actually, following Rogers (1970), both Ho and lo arise from the same construc-tion, just changing the initial class S (see ‘‘Method II’’, Theorem 15, Chap. 1). Ittrivially follows that

loðAÞ � HoðAÞ for all A � Rn;

and being interested in null sets, in the rest of the paper we shall confine ourselves togive estimates for the spherical Hausdorff measure Ho. Moreover, we shall abbre-viate the terminology and by Hausdorff measure we shall actually mean sphericalHausdorff measure. For further notions about this kind of measures the reader isreferred to the classical treatise (Rogers, 1970) where Hausdorff measures have beenintroduced and studied using general functions as o above (see Hausdorff, 1918 forthe historically first definition, Rogers, 1970 and related references).

The Singular Set. To be more precise concerning the regularity of Du, we saythat x0 2 O is an interior regular point for Du iff Du is continuous in a neighborhoodof x0; the set of interior regular points of u is denoted by Reg u. Then we can definethe singular set of a weak solution u 2 W1;2ðO;RN Þ \ L1 to (1.7) as follows:

Sing u :¼ fx 2 O : x is not a regular point of Dug ¼ OnReg u:

For a weak solution to an elliptic system of the type in (1.7), under the assumptionsstated above, it is known that there exists an open subset O0 � O such that

Elliptic Systems, Singular Sets and Dini Continuity 1219

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Du 2 C0locðO0;MN�nÞ, i.e., O0 � Reg u, and moreover that HnðOnO0Þ ¼ jOnO0j ¼ 0.

Therefore jSing uj ¼ 0.

Results. The first theorem we state regards solutions to systems of the type(1.7).

Theorem 2.1. Let u 2 W1;2ðO;RN Þ \ L1ðO;RN Þ be a weak solution to the system(1.7), under the assumptions (2.1), (2.2), and (2.5). There exists a numbert � tðn;N ;L; n;C; kuk1;dÞ 2 ð0; 1Þ, independent of the vector fields a and b, ofthe solution u and of the function mðÞ, such that if

LðrÞ :¼Z r

0

mðrÞtr

dr < 1 ð2:7Þ

then for any e > 0

HoðSing uÞ ¼ 0; where oðrÞ ¼ rnLðrÞe�3:

The presence of the factor t in (2.7) leads us to assume the Dini continuity of thefunction mt, which is a stronger assumption than the one prescribing the Dini conti-nuity of the functions m, since t can be very small. On the other hand, exactly for thesame reason, bounds from below for t can be explicitly given in terms of n, L and d.Indeed, t essentially depends on the higher integrability exponent d0 fromTheorem 2.3 and in turn on the higher integrability constant of Gehring’s lemma,for which bounds from below are available (see Bojarski and Iwaniec, 1983). There-fore, (2.7) is effective. We note that the presence of a small exponent t in (2.7) is notcaused by the Dini continuity assumption. It is an effect of the explicit dependence ofthe vector field a on u. Indeed, the effect already shows up in the case where a isHolder or Lipschitz continuous with respect to x and u, i.e., mðrÞ ¼ ra. In that casethe right-hand side in the estimate (1.5) of the singular set must be replaced by n� s,where s is a small positive number exhibiting the same dependencies as t (seeMingione, 2003b, Theorem 2.5). As a matter of fact, Theorem 2.1 contains this lastresult in the case mðrÞ ¼ ra.

When no direct dependence of the coefficient a on u is allowed, Theorem 2.1 canbe improved as follows:

Theorem 2.2. Let u 2 W1;2ðO;RN Þ \ L1ðO;RN Þ be a weak solution to the system

�div aðx;DuÞ ¼ bðx; u;DuÞ ð2:8Þ

under the assumptions (2.1)–(2.3), and (2.4). There exists an exponentt ¼ tðn;N ;L; n;C;dÞ 2 ð1;1Þ, independent of the vector fields a and b, of thesolution u and of the function mðÞ, such that if

LðrÞ :¼Z r

0

mðrÞ23tr

dr < 1 ð2:9Þ

1220 Duzaar, Gastel, and Mingione

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then for any e > 0 we have

HoðSing uÞ ¼ 0; where oðrÞ ¼ rnLðrÞe�3t:

In the case mðrÞ ¼ ra, the theorem recaptures the result of the estimate in (1.5),connecting the Hausdorff dimension of the singular set with the modulus of continuityof the coefficients. We note that the occurrence of a function of the type LðrÞ in the esti-mate of the Hausdorff dimension of the singular set is natural in view of the fact that weneed a similar function as the modulus of continuity of Du around regular points (see(1.6)). It is not clear to us how to drop the factor 2

3 from (2.9). Finally, we note thatthe use ofHausdorff measures defined in terms of functions which are not powers is typi-cal when estimating the singular set in borderline cases, see for instance Choe and Lewis(2000).

Preliminary Material. We conclude this section with a few preliminary resultswe shall need later. For a proof of the next theorem, see Giaquinta and Modica(1979, Proposition 2.1).

Theorem 2.3. Let u 2 W1;2ðO;RN Þ \ L1ðO;RN Þ be a weak solution of the ellipticsystem (1.1), under the assumptions (2.1), (2.2), and (2.5). Then there exists a

constant d0 ¼ d0ðn;N ;L; n;dÞ 2 ð0;1Þ, such that Du 2 L2ð1þd0Þloc ðO;MN�nÞ:

More precisely, the following reverse-Holder inequality is valid for any ball BR YO:��ZBgR

jDuj2ð1þd0Þ dx� 1

ð1þd0Þ � ~cc�Z

BR

ð1þ jDuj2Þdx ð2:10Þ

where ~cc � ~ccðn;N ;L; n;d; gÞ and g 2 ð1=2; 1Þ. Another consequence of the previoustheorem is that whenever eOOYO is an open subset, there exists a constantc ¼ cðdistðeOO; @OÞÞ (also depending on n, N , L, n, and d) such that:

kDukL2ð1þd0Þð~OO;MN�nÞ � ckDukL2ðO;MN�nÞ: ð2:11Þ

Note that the previous results are only based on monotonicity and do not require anyspecial continuity assumption on aðx; u;DuÞ with respect to ðx; uÞ.

For a vector valued function G : Rn ! Rk, a unit vector e 2 Rn and a realnumber h 2 R, we define

te;hGðxÞ ¼ Gðxþ heÞ �GðxÞ:The following Lemma is a straightforward consequence of te;hGðxÞ ¼R h

0 DeGðxþ teÞdt.

Lemma 2.4. If 0 < r < R, 0 < jhj < R� r, 1 � q < 1, and G 2 LqðBRÞ, then(trivially)Z

Br

jGðxþ heÞjq ds �ZBR

jGðxÞjq dx

Elliptic Systems, Singular Sets and Dini Continuity 1221

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moreover, if DeG 2 LqðBRÞ thenZBr

jte;hGðxÞjq dx � jhjqZBR

jDeGðxÞjq dx:

The following formulation of the classical Gehring’s lemma is suitable for ourpurposes later in Chap. 4 (see Stredulinsky, 1980).

Lemma 2.5. Let fvhgh>0 � L2locðO;RN Þ, fwhgh>0 � L

2ð1þd1Þloc ðO;RN Þ be two families

of functions such that

�Z

BR=2

jvhj2 dx � c

��Z

BR

jvhj2s dx�1

s

þ c �Z

BR

jwhj2 dx;

for any BR YO, where c; c 2 ½1;1Þ, d1 > 0, and s 2 ð0; 1Þ are independent of h.Then there exist ~cc ¼ ~ccðn;N ; s; c; d1Þ and 1 < b ¼ bðn;N ; s; c; d1Þ < ð1þ d1Þ indepen-dent of h and c, such that for any BR YO we have:

��Z

BR=2

jvhj2b dx�1

b

� ~cc�Z

BR

jvhj2 dxþ ~ccc

��Z

BR

jwhj2ð1þd1Þ dx� 1

1þd1

:

For a proof of the following comparison theorem we refer to Rogers (1970,Theorem 40) .

Theorem 2.6. Let f ; g : ½0;1Þ ! ½0;1Þ be increasing continuous functions withfð0Þ ¼ gð0Þ ¼ 0 and such that

limr&0

fðrÞgðrÞ ¼ 0:

Then for any A � Rn it follows:

HgðAÞ < 1¼)Hf ðAÞ ¼ 0:

3. A VARIANT OF FRACTIONAL SOBOLEV SPACES

In the following, by a modulus of continuity we understand a continuous,increasing function ~mm : ½0;1Þ ! ½0;1Þ such that ~mmð0Þ ¼ 0. For later purposes, weintroduce functions which belong to a certain type of limiting fractional Sobolevspace; we define W ~mm;qðO;RN Þ to be the space of measurable functions u : O ! RN

for which

kukW ~mm;q :¼ kukLq þ�Z

O

ZO

juðxÞ � uðyÞjqjx� yjn~mmðjx� yjÞq dx dy

�1q

< 1: ð3:1Þ

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We note that the previous quantity is a norm making W ~mm;qðO;RN Þ a Banach space.The local variant, i.e., W ~mm;q

loc ðO;RN Þ, can be defined as usual. Observe that whenmðrÞ � ra the previous choice gives back the usual definition of the fractionalSobolev space Wa;q

loc ðO;RN Þ. Actually we shall need no function space theoretic prop-erty of such spaces. We could have even avoided introducing them; we did so inorder to have a bit of notation at our disposal; the only thing we actually need isto know that functions satisfying (3.1) also satisfies Proposition 3.1 below and thatevery function satisfying the assumption of Proposition 3.3 fulfills also (3.1) forsuitable moduli of continuity.

The following is a simple Poincare type inequality for functions belonging toW

~mm;qloc ðO;RN Þ:

Proposition 3.1. Let G 2 W~mm;qloc ðO;RN Þ for some q 2 ½1;1Þ and Brðx0ÞYO, and let

~mm be such that ~mmð2rÞ � A~mmðrÞ, thenZBrðx0Þ

jGðxÞ � ðGÞx0;rjqdx

� 2nAqo�1n ~mmðrÞq

ZBrðx0Þ

ZBrðx0Þ

jGðxÞ �GðyÞjqjx� yjn~mmðjx� yjÞq dx dy:

Proof. Fix x 2 Brðx0Þ. Then, by Jensen’s inequality, we have

jGðxÞ � ðGÞx0;rjq � �

ZBrðx0Þ

jGðxÞ �GðyÞjq dy

� �Z

Brðx0ÞjGðxÞ �GðyÞjqKeðjx� yjÞdy

where, for e 2 ð0; 1Þ, KeðtÞ :¼ minfð2r=tÞnð~mmð2rÞ=~mmðtÞÞq; e�1g (observe that weused the fact that KeðtÞ � 1 whenever t � 2r). Integrating with respect to x, wededuceZ

Brðx0ÞjGðxÞ � ðGÞx0;rj

qdx �

ZBrðx0Þ

�Z

Brðx0ÞjGðxÞ �GðyÞjqKeðjx� yjÞdx dy

and the statement follows letting e& 0. The constant in the inequality finally followskeeping into account that ~mmð2rÞ � A~mmðrÞ. &

We note that this last inequality is trivially satisfied with A ¼ 2 if r 7! ~mmðrÞ isconcave.

Remark 3.2. If G 2 W ~mm;qðO;RN Þ then the Poincare type inequality stated inProposition 3.1 holds for any ball Brðx0Þ � O.

The following result generalizes a well known characterization of fractionalSobolev spaces via Nikolskii spaces, see Nikol’skii (1975, Chap. 6).

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Proposition 3.3. Let G 2 LqðB3R;MN�nÞ for some q 2 ½1;1Þ, and assume thereexist M � 0, A � 1, and a modulus of continuity ~mm satisfyingZ 1

0

~mmðrÞqr

dr < 1

such thatZSn�1

ZB2R

jte;hGðxÞjq dx dHn�1ðeÞ � Mq~mmðhÞq

for every h with h < RA. Then, G 2 Wy;qðBR;MN�nÞ for every modulus of continuity y

satisfying the relative Dini conditionZ 1

0

~mmðrÞqryðrÞq dr < 1: ð3:2Þ

Moreover, there exists a constant c ¼ cðn;R;A; ~mmð Þ; yð ÞÞ such that

kGkWy;qðBRÞ � cðM þ kGkLqðB3RÞÞ:

Proof. Let r :¼ R2A. The assumption impliesZ

Sn�1

ZBrðx0Þ

jte;hGðxÞjq dx dHn�1ðeÞ � Mq~mmðhÞq

for every h < RAand every x0 2 BR. Using the co-area formula and Fubini’s theorem,

with ShðyÞ :¼ fx 2 RN : jx� yj ¼ hg, we obtainZBrðx0Þ

ZBrðx0Þ

jGðxÞ �GðyÞjqjx� yjnyðjx� yjÞq dx dy

¼ZBrðx0Þ

Z 2r

0

ZShðyÞ\Brðx0Þ

jGðxÞ �GðyÞjqhnyðhÞq dHn�1ðxÞdhdy

�ZBrðx0Þ

Z 2r

0

ZSn�1

jGðyþ heÞ �GðyÞjqhyðhÞq dHn�1ðeÞdhdy

¼Z 2r

0

ZSn�1

ZBrðx0Þ

jGðyþ heÞ �GðyÞjqhyðhÞq dy dh dHn�1ðeÞ

�Z 2r

0

�ZSn�1

ZBrðx0Þ

jte;hGðyÞjq dy dHn�1ðeÞ�

dh

hyðhÞq

� Mq

Z 2r

0

~mmðrÞqryðrÞq dr

� Mq

Z R

0

~mmðrÞqryðrÞq dr

� cðR; ~mmð Þ; yð ÞÞMq:

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Now we cover BR by cðnÞAn balls Br=2ðxiÞ, xi 2 BR, and observe that for anyðx; yÞ 2 Z,

Z :¼�ðx; yÞ 2 BR � BR : jx� yj � R

4A

�;

there is some i such that x; y 2 BrðxiÞ. Hence the above chain of inequalitiesimpliesZ

Z

jGðxÞ �GðyÞjqjx� yjnyðjx� yjÞq dðx; yÞ � cðnÞAncðR; ~mmð Þ; yð ÞÞMq;

while on the other handZðBR�BRÞnZ

jGðxÞ �GðyÞjqjx� yjnyðjx� yjÞq dðx; yÞ �

�R

4A

��n

y�

R

4A

��q

�ZB3R�B3R

jGðxÞ �GðyÞjq dðx; yÞ � cðnÞAny�

R

4A

��q ZB3R

jGðxÞjq dx:

A combination of the last two estimates proves the proposition. &

4. ‘‘SUB-FRACTIONAL’’ ESTIMATES

In this section, we derive some ‘‘sub-fractional’’ estimates we shall need in thesequel. Our starting point are estimates somehow similar to those in Mingione(2003a,b); anyway, in many points, we shall argue in a completely different way toprove the estimates we need.

In the weak formulation,ZOað; u;DuÞDjdx ¼

ZObð; u;DuÞjdx ð4:1Þ

valid for any j 2 W1;20 ðO;RN Þ \ L1ðO;RN Þ, we fix a ball BRðx0Þ � BR YO and a

unit vector e 2 Rn. We then choose

j :¼ te;�hðZ2te;huÞ

as test-function in (4.1) where 0 < h � R1000 and Z 2 C1

0 ðB3R=4Þ is a cut-off functionsatisfying Z � 1 on BR=2, 0 � Z � 1, and jDZj � 8

R. We obtainZ

OZ2te;h

�að; u;DuÞ�te;hDu dx

¼ �ZO2Z te;h

�að; u;DuÞ�DZ� te;hu dxþ

ZObð; u;DuÞt�e;h

�Z2te;hu

�dx:

ð4:2Þ

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Now we write

te;hðaðx; uðxÞ;DuðxÞÞÞ¼ aðxþ he; uðxþ heÞ;Duðxþ heÞÞ � aðx; uðxþ heÞ;Duðxþ heÞÞþ aðx; uðxþ heÞ;Duðxþ heÞÞ � aðx; uðxÞ;Duðxþ heÞÞþ aðx; uðxÞ;Duðxþ heÞÞ � aðx; uðxÞ;DuðxÞÞ

¼: AðhÞ þBðhÞ þ CðhÞ:

with the obvious labeling. With this notation, (4.2) turns intoZOZ2�AðhÞ þBðhÞ þ CðhÞ�te;hDu dx

¼ �ZO2Z�AðhÞ þBðhÞ þ CðhÞ�DZ� te;hu dx

þZObð; u;DuÞt�e;h

�Z2te;hu

�dx: ð4:3Þ

We proceed estimating the various integrals arising in (4.3).Using (2.1)3, Young’s inequality and Lemma 2.4, the first term of the left-hand

side of (4.3) can be estimated from above as follows:ZOZ2jAðhÞte;hDujdx

� LmðhÞZOZ2�1þ jDuðxþ heÞj�jte;hDujdx

� eZOZ2jte;hDuj2 dxþ 1

eL2mðhÞ2

ZOZ2�1þ jDuðxþ heÞj2�dx

� eZOZ2jte;hDuj2 dxþ 1

eL2mðhÞ2

ZBR

�1þ jDuj2�dx :

Using again (2.1)3, Lemma 2.4 (recall that supp ZYB3R=4 and h � R1000), and Cauchy–

Schwarz inequality, we obtain for the first integral of the right-hand side of (4.3):ZO2ZjAðhÞDZ� te;hujdx

� 2LmðhÞZOZjDZj�1þ jDuðxþ heÞj�jte;hujdx

� 4LkDZk1mðhÞ�Z

OZ�1þ jDuðxþ heÞj2�dx�1

2�Z

B3R=4

jte;huj2 dx�1

2

� c mðhÞhZBR

�1þ jDuj2�dx

� c mðhÞ2ZBR

�1þ jDuj2�dx;

1226 Duzaar, Gastel, and Mingione

ORDER REPRINTS

where we have used (2.4) in the the last line. Note that the constant c is given by4LkDZk1.

Now we turn our attention to the second integral appearing on the left-hand sideof (4.3). Using again (2.1)3 together with the fact that u 2 L1ðO;RN Þ and Young’sinequality with e 2 ð0; 1Þ, we infer

ZOZ2jBðhÞte;hDujdx

� L

ZOZ2�1þ jDuðxþ heÞj�jte;hDuj mðjte;hujÞdx

� eZOZ2jte;hDuj2 dxþ 1

eL2

ZOZ2�1þ jDuðxþ heÞj2�mðjte;hujÞ2 dx:

To estimate the second integral appearing on the right-hand side of (4.3), we firstobserve that we have the estimate jte;huj � cðkuk1Þmðjte;hujÞ. This estimate caneasily be derived from the fact that u is bounded: In the case thatjte;huðxÞj � 1 the estimate follows directly from (2.4); in the remaining casejte;huðxÞj > 1, we observe jte;huðxÞj � 2kuk1 � 2kuk1mðjte;huðxÞjÞ; here we haveused the fact that r 7! mðrÞ is non-decreasing together with the normalizationmð1Þ ¼ 1. This proves the desired estimate with c ¼ max 1; 2kuk1

. Combining

it with (2.1)2, we deduce

ZO2ZjBðhÞDZ� te;hujdx � c

ZOZ�1þ jDuðxþ heÞj2�mðjte;hujÞ2 dx;

where the constant c is given by 4Lmax 1; 2kuk1 kDZk1.

To treat the integrals involving the quantity CðhÞ, we rewrite CðhÞ, asfollows:

CðhÞ ¼Z 1

0

Dza�; u;Duþ tte;hDu

�dt te;hDu ¼: eCCðhÞte;hDu: ð4:4Þ

Recalling the growth and ellipticity assumptions from (2.1) we see that

njte;hDuj2 � eCCðhÞte;hDu� te;hDu and jeCCðhÞj � L: ð4:5Þ

Hence the third integral on the left-hand side of (4.3) can be estimated from below asfollows:

ZOZ2CðhÞte;hDu dx � n

ZOZ2jte;hDuj2 dx: ð4:6Þ

Elliptic Systems, Singular Sets and Dini Continuity 1227

ORDER REPRINTS

Using (4.4), Young’s inequality with e 2 ð0; 1Þ, the bound jeCCðhÞj � L from (4.5), andLemma 2.4 we obtainZ

O2Z jCðhÞDZ� te;hujdx

¼ZO2Z��eCCðhÞte;hDu� ðDZ� te;huÞ

��dx� e

ZOZ2jte;hDuj2 dxþ 1

eL2kDZk21

ZB3R=4

jte;huj2 dx

� eZOZ2jte;hDuj2dxþ 1

eL2kDZk21h2

ZBR

�1þ jDuj2�dx

� eZOZ2jte;hDuj2 dxþ 1

eL2kDZk21mðhÞ2

ZBR

�1þ jDuj2�dx;

where we have used once again (2.4).We finally choose e ¼ n

6. Then, combining the previous estimates, we obtain from(4.3)Z

OZ2jte;hDuðxÞj2 dx � c1mðhÞ2

ZBR

�1þ jDuj2�dx

þ c2

ZOZ�1þ jDuðxþ heÞj2�mðjte;hujÞ2 dxþ 6C

n

ZO

�1þ jDuj2�jte;�hðZ2te;huÞjdx;

ð4:7Þ

whenever Z is a cut-off function and

BR YO Z 2 C10 ðB3R=4Þ Z � 1 on BR=2 jDZj � 8

R0 < h � R

1000:

Here c1 ¼ c1�Ln ; kDZk1

�and c2 ¼ c2

�Ln ; kDZk1; kuk1

�are constants independent of

the considered ball BR YO and the displacement h.We now turn our attention to the third integral appearing on the right-hand side

of (4.7). Using the fact that the support Zð � heÞ is contained in B7R=8, Holder’sinequality, Theorem 2.3, i.e., (2.10) (with g :¼ 7=8) and Lemma 2.4, we deriveZ

Oð1þ jDuj2Þjte;�hðZ2te;huÞjdx

�ZB7R=8

�1þ jDuðxÞj2�jðte;huÞðx� heÞjdxþ

ZB7R=8

�1þ jDuðxÞj2�jðte;huÞðxÞjdx

� 2

�ZB7R=8

�1þ jDuj2ð1þd0Þ�dx� 1

1þd0�Z

B7R=8

jðte;huÞðx� heÞj1þd0d0 dx

� d01þd0

þ 2

�ZB7R=8

�1þ jDuj2ð1þd0Þ�dx� 1

1þd0�Z

B7R=8

jðte;huÞðxÞj1þd0d0 dx

� d01þd0

1228 Duzaar, Gastel, and Mingione

ORDER REPRINTS

� cðRÞkuk1

1þd01ZBR

�1þ jDuj2�dx�Z

B7R=8

jðte;huÞðxÞj þ jðte;�huÞðxÞjdx� d0

1þd0

� cðRÞkuk1

1þd01ZBR

�1þ jDuj2�dx�Z

BR

ð1þ jDujÞdx� d0

1þd0

hd0

1þd0

� cðRÞkuk1

1þd01

�ZBR

�1þ jDuj2�dx�1þ2d0

1þd0

hd0

1þd0

¼ c mðhÞd0

1þd0 ð4:8Þ

where the constant c depends on n, N , L, n, C, d, kuk1, kDuk2, and R. (We note thatthe constant cðRÞ appearing in the above chain of inequalities has the same depen-dencies as the constant in the estimate (2.10).) We also note that in a crucial way wehave used the fact that u is bounded; moreover, since we have used the higherintegrability estimate (2.10), the constant in the last line of the previous estimatedepends on the difference d appearing in (2.6) via the higher integrability exponentd0 from Theorem 2.3, in the sense that c ! 1 when d & 0.

It remains to bound the second integral appearing on the right-hand side of(4.7). We proceed as before, using the concavity of the function t 7! mðtÞ and Jensen’sinequality, to obtainZ

OZ�1þ jDuðxþ heÞj2�mðjte;hujÞ2 dx

� 2

�ZB3R=4

�1þ jDuðxþ heÞj2ð1þd0Þ�dx� 1

1þd0�Z

B3R=4

m1þd0d0 ðjte;hujÞdx

� d01þd0

� 2m�2kuk1

� 11þd0

�ZB7R=8

�1þ jDuj2ð1þd0Þ�dx� 1

1þd0�Z

B3R=4

mðjte;hujÞdx� d0

1þd0

� c

��Z

B3R=4

mðjte;hujÞdx� d0

1þd0 � c md0

1þd0

��Z

B3R=4

jte;hujdx�

� c md0

1þd0

�cðnÞh�

ZBR

jDujdx�

� c md0

1þd0ðhÞ;

where the constant c exhibits the same dependence on the parameters as the one in(4.8). We note that we have used the concavity of m to perform the last estimate.

Combining the last and the second last estimate with (4.7) and lettingb :¼ d0

2ð1þd0Þ, we obtain the following:

Proposition 4.1. Suppose u 2 W1;2ðO;RN Þ \ L1ðO;RN Þ is a weak solution of theelliptic system (1.7) such that the hypotheses (2.1)–(2.5) are fulfilled. Then forany BR � O and any unit vector e 2 Rn the inequalityZ

BR=2

jte;hDuðxÞj2 dx � cm2bðhÞ 0 < h � R

1000; ð4:9Þ

Elliptic Systems, Singular Sets and Dini Continuity 1229

ORDER REPRINTS

holds with b ¼ bðn;N ;L; n;C; kuk1;dÞ 2 ð0; 1Þ and c ¼ cðn;N ;L; n;C; kuk1;kDuk2;d;RÞ.

We note that the exponent b depends critically on d, via d0: indeed b ! 0 whend & 0.

Now we want to derive a similar estimate for elliptic systems of the type (2.8),i.e., systems in which the vector field a do not depend on u; the relevant differenceis that in this case we can allow the exponent b to be strictly larger than 1. In thecase of systems which are of the type (2.8) the term BðhÞ does not occur in (4.3),and therefore (4.7) reduces toZ

OZ2jte;hDuðxÞj2 dx

� c mðhÞ2ZBR

�1þ jDuj2�dxþ 6C

n

ZO

�1þ jDuj2�jte;�hðZ2te;huÞjdx ; ð4:10Þ

where the constant c depends only on Ln and kDZk1. The starting point of our

considerations is the following estimate:

ZBR2

jte;hDuj2 dx � cm2ðhÞ 0 < h � R

1000; ð4:11Þ

where c ¼ cðn;N ;L; n;C;d; kuk1; kDuk2; mðÞ;RÞ. This estimate is easily derived asfollows:

Using (2.3) with a ¼ d02ð1þd0Þ, i.e., h

d01þd0 � Aða; mðÞÞ2mðhÞ2, the second integral

appearing on the right-hand side of (4.10) is estimated from above by2Cn cAða; mðÞÞ2mðhÞ2, where c denotes the constant from (4.8). Inserting this into(4.10) yields the desired estimate.

In the following eOOYO is a fixed open set and 0 < h � 14 dist ðeOO; @OÞ. It is

worth noting here that, in the following estimates, and more precisely up to(4.22) below, we will not assume that h does not exceed R or any multiple of R;the only bound for h is the one given above in terms of O and eOO. Therefore wewill not use the second inequality from Lemma 2.4. What we want to do insteadis finding estimates which are uniform with respect to h. The reason for this is thatwe want to apply Gehring’s lemma, i.e., Lemma 2.5, to the scaled functionsvh ¼ mðhÞ�1te;hu on eOO and therefore we have to consider the balls BR Y eOO indepen-dently of h.

Let e be a fixed but arbitrary unit vector in Rn and j 2 C10 ðeOO;RN Þ. Then we

take te;�hj as test function in the weak formulation (4.1). According to the notationintroduced at the beginning of the section we obtain, via ‘‘integration by parts’’ forfinite differencesZ

eO �AðhÞ þ CðhÞ�Djdx ¼ZeR bð; u;DuÞte;�hjdx : ð4:12Þ

1230 Duzaar, Gastel, and Mingione

ORDER REPRINTS

We define

vh :¼ te;humðhÞ

fAAðhÞ :¼ �AðhÞmðhÞ : ð4:13Þ

Dividing (4.12) by mðhÞ, we arrive atZeO eCCðhÞDvh �Djdx ¼

ZeO½fAAðhÞDjþ mðhÞ�1

bð; u;DuÞte;�hj�dx ð4:14Þ

which is valid for any j 2 C10 ðeOO;RN Þ. The definition of eCCðhÞ from (4.4) implies that

vh 2 W1;2ðeOO;RN Þ is a weak solution of the linear system (4.14), which is elliptic withellipticity constant n and upper bound L; i.e., from (1.2) we infer that

njxj2 � eCCðhÞx� x and eCCðhÞx� ~xx � Ljxjj~xxj ; ð4:15Þ

holds whenever x; ~xx 2 MN�n and h is as above. For later use we note that (2.1)3implies the bound

jfAAðhÞj � Lð1þ jDuðxþ heÞjÞ : ð4:16Þ

In the sequel we use the abbreviation v � vh, recovering the full notation if necessary.We fix 0 < h � 1

4 distðeOO; @OÞ as above and consider an arbitrary (but fixed) ballB2R Y eOO. By Z 2 C1

0 ðB3R=4Þ we denote an associated cutoff function such thatZ � 1 on BR=2 and jDZj � 8

R. We then use j :¼ Z2ðv� ðvÞRÞ as test function in

(4.14). Taking into account the identity Dj ¼ Z2Dvþ 2ZDZ� ðv� ðvÞRÞ, the left-hand side of (4.14) splits into two parts. The first one is estimated by the use ofthe ellipticity bound in (4.15) as follows:Z

BR

Z2eCCðhÞDv�Dvdx � nZBR

Z2jDvj2 dx : ð4:17Þ

Using the upper bound in (4.15) and Young’s inequality with e 2 ð0; 1Þ, we get forthe second integral appearing on the left-hand side of (4.14):Z

BR

2Z jeCCðhÞDv� ðDZ� ðv� ðvÞRÞÞj dx

� eZBR

Z2jDvj2 dxþ 64L2

eR2

ZBR

jv� ðvÞRj2 dx: ð4:18Þ

Similarly, applying Young’s inequality twice and taking (4.16) into account, we see thatZBR

jfAAðhÞDjjdx � eZBR

Z2jDvj2 dxþ 64L2

eR2

ZBR

jv� ðvÞRj2 dx

þ�L2

eþ e�Z

BR

�1þ jDuðxþ heÞj�2 dx : ð4:19Þ

Elliptic Systems, Singular Sets and Dini Continuity 1231

ORDER REPRINTS

As far as the right hand side is concerned we obtain, using Young’s inequality withexponents 1þ d0

2 and 1þ 2d0, the boundedness of u, a variant of Lemma 2.4, (2.3) in

the case a ¼ d02ð2þd0Þ, and Poincare’s inequality:Z

BR

mðhÞ�1jbð; u;DuÞjjte;�hðZ2ðv� ðvÞRÞÞjdx

� C

�ZBR

�1þ jDuj�2þd0

dxþ mðhÞ�2ð2þd0Þ

d0

ZBR

jte;�hðZ2ðu� ðuÞRÞÞj1þ2d0 dx

� cðC; kuk1; d0Þ

�ZBR

�1þ jDuj�2þd0

dxþ hmðhÞ�2ð2þd0Þ

d0

ZBR

jDðZ2ðu� ðuÞRÞÞjdx

� cðC; kuk1; d0Þ�Z

BR

�1þ jDuj�2þd0

dxþAða;mðÞÞ1aZBR

jDðZ2ðu� ðuÞRÞÞjdx

� cðC; kuk1; d0;mðÞÞ�Z

BR

�1þ jDuj�2þd0

dx

þ 1

R

ZBR

ju� ðuÞRjdxþZBR

jDujdx

� c

ZBR

�1þ jDuj�2þd0

dx ; ð4:20Þ

where c ¼ cðn;N ;L; n;C;d; kuk1;mðÞÞ. We note that the dependence of the constant con d appears sincewe have used the higher integrability of uwith exponent 2þ d0; cf. theremark just after the estimate (4.8). The variant of Lemma 2.4 we alluded to is thefollowing (recall that we are not supposing h � R here):Z

BR

jte;�hðZ2ðu� ðuÞRÞÞjdx � h

ZOjDðZ2ðu� ðuÞRÞÞjdx :

This can be shown by using Fubini’s theorem and the fact that supp ZYBR (whichallows us to replace O by BR as domain of integration in the right-hand side).

Inserting the estimates (4.17)–(4.20) in (4.14) and choosing e ¼ n4, we finally

arrive at the following Caccioppoli type estimate:ZBR=2

jDvhj2 dx � c

R2

ZBR

jvh � ðvhÞRj2 dxþ c3

�ZBR

�1þ jDuðxÞj þ jDuðxþ heÞj�2þd0

dx : ð4:21Þ

where c ¼ c�Ln

�and c3 ¼ c3ðn;N ;L; n;C;d; kuk1; mðÞÞ. Therefore, using the

Sobolev–Poincare inequality, we obtain

�Z

BR=2

jDvhj2 dx � c

��Z

BR

jDvhj2nnþ2 dx

�nþ2n

þ c3

� �Z

BR

ð1þ jDuðxÞj þ jDuðxþ heÞjÞ2þd0dx; ð4:22Þ

1232 Duzaar, Gastel, and Mingione

ORDER REPRINTS

where c ¼ c�n; Ln

�and c3 is as before. Note that c and c3 are independent of h. Now,

look at (2.11) and observe that�1þ jDuj þ jDuð þ heÞj�2þd0 2 L

2þ2d02þd0 ðeOOÞ :

Since BR Y eOO is arbitrary and the previous inequality is uniform with respect to h, wecan apply Gehring’s Lemma 2.5 with O replaced by eOO, s :¼ n

nþ2 and

1þ d1 :¼ 2þ2d02þd0

> 1. We obtain with a fixed exponent bðn;N ;L; n;dÞ 2 ð1;1Þ and a

constant ~cc ¼ cðn;N ;L; n;dÞ < 1, both independent of h, for any BR Y eOO:��Z

BR=2

jDvhj2b dx�1

b

� ~cc�Z

BR

jDvhj2 dxþ ~ccc3

��Z

BR

�1þ jDuðxÞj þ jDuðxþ heÞj�2þ2d0

dx

� 2þd02þ2d0

:

ð4:23Þ

At this stage, we fix a ball B2R Y eOO and take h such that 0 < h � R1000. Using (4.11)

and the definition of vh from (4.13) we see that the first integral of the right-hand sidein the previous inequality is uniformly bounded by a constant c having the samedependencies as the constant appearing in (4.11). Moreover, the second integralon the right-hand side of (4.23) can be estimated by cðRÞjjDujj2þd0

L2ðB2RÞ via (2.10) andLemma 2.4. This proves the following:

Proposition 4.2. Let u 2 W1;2ðO;RN Þ \ L1ðO;RN Þ be a weak solution of the ellip-tic system (2.8) such that the assumptions (2.1)–(2.5) are satisfied. Then for anyB2R YO and any unit vector e 2 Rn the inequalityZ

BR=2

jte;hDuj2b dx � cm2bðhÞ ; 0 < h � min

�R

1000;1

4distðB2R; @OÞ

�ð4:24Þ

holds with an exponent b ¼ bðn;N ;L; n;dÞ 2 ð1;1Þ and a constant c ¼cðn;N ;L; n;C;d; kuk1; kDuk2; mðÞ;RÞ.

We remark that also in this case the exponent b depends critically on d in thesense that b&1 when d&1. Also observe the following: in order to assert thatthe exponent b is independent of mðÞ we need the full version of Lemma 2.5, statingthat the constant c does not affect the higher integrability exponent. In the estimatesabove, the only point where an intrinsic dependence on mðÞ comes into the play is(4.20), where we use (2.3). This determines c3 that we treat via Lemma 2.5 as c.

5. DIMENSION REDUCTION

We begin this section with a generalization of Giusti’s Lemma (see Giusti, 1969).Here o : ½0;1Þ ! ½0;1Þ stands for an increasing, continuous function satisfying

Elliptic Systems, Singular Sets and Dini Continuity 1233

ORDER REPRINTS

oð0Þ ¼ 0, and Ho denotes the generalized spherical Hausdorff measureassociated to o.

Lemma 5.1. Let A 2 RN be an open subset and l a finite, non-negative, increasing,finite set function on the family of all open subsets of A, which is countably super-additive in the sense that

l

[1i¼1

Oi

!�Xni¼1

lðOiÞ

whenever ðOiÞi2N is a countable family of disjoint open subsets of A, and whichsatisfies

lðOÞ ! 0 whenever O � A is open with jOj ! 0: ð5:1ÞMoreover, assume that

ð0;1Þ 3 r 7! rn

oðrÞ is non-decreasing ð5:2Þ

limr&0

rn

oðrÞ ¼ 0: ð5:3Þ

Then HoðEÞ ¼ 0, where

E ¼�y 2 A : lim sup

r&0

oðrÞ�1lðBrðyÞÞ > 0

�:

Proof. For j 2 N, we define Ej ¼y 2 A : lim supr&0 oðrÞ�1lðBrðyÞÞ > 1

j

. Let K

be a compact subset of A and 0 < e < distðK; @AÞ. For y 2 Ej \ K, we know thatthere exists a radius rðyÞ 2 ð0; eÞ such that

oðrðyÞÞ�1lðBrðyÞðyÞÞ �1

2j:

In this situation, we can pick a maximal pairwise disjoint countable collection ofballs

�BriðyiÞ

�i2N with yi 2 Ej \ K, ri ¼ rðyiÞ such that

�B5riðyiÞ

�i2N covers Ej \ K.

Using the estimate from above with ri, yi instead of rðyÞ, y and summing overi 2 N, we then have

X1i¼1

oðriÞ � 2jX1i¼1

lðBriðyiÞÞ � 2jl

[1i¼1

BriðyiÞ!

� 2jlðAÞ :

Moreover,�����[1i¼1

BriðyiÞ����� ¼ an

X1i¼1

rni � anen

oðeÞX1i¼1

oðriÞ � 2janen

oðeÞ lðAÞ:

1234 Duzaar, Gastel, and Mingione

ORDER REPRINTS

Here, we have used (5.2) which also implies that oð5rÞ � 5noðrÞ for r > 0.Combining this with the previous estimate, we see that

X1i¼1

oð5riÞ � 5nX1i¼1

oðriÞ � 5n2jl

[1i¼1

BriðyiÞ!:

Recalling (5.3), we conclude that��S1

i¼1 BriðyiÞ��! 0 as e ! 0. Combining this with

(5.1), we obtain that HoðEj \ KÞ ¼ 0. Since j 2 N is arbitrary, this proves theclaim. &

From Duzaar and Gastel (2002, Theorem 1.2) we recall that ifu 2 W1;2ðO;RnÞ \ L1 is any weak solution to (1.7) such that 2Ckuk1 < n then thefollowing inclusion for the singular set of u holds:

Sing u � S0 [ S1

where

S0 ¼(y 2 O : lim inf

r&0�Z

BrðyÞjDu� ðDuÞy;rj2 dx > 0

);

and

S1 ¼�y 2 O : lim sup

r&0

jðDuÞy;rj ¼ 1�:

Therefore, when proving Theorems 2.1 and 2.2, we shall estimate the size of S0 andS1 separately. We start with the proof of Theorem 2.2 which is a bit more delicate.Then, we give the adjustments to achieve the proof of Theorem 2.1.

Proof of Theorem 2:2. Estimate for S0. In the following b ¼ bðn;N ;L; n;dÞ 2ð1;1Þ denotes the exponent from Proposition 4.2. From Proposition 4.2,Proposition 3.3 (here we take ~mm ¼ m and q ¼ 2b), and a standard coveringargument we infer that Du 2 W

y;2bloc ðO;MN�nÞ where yðÞ is any modulus of continu-

ity satisfying

Z 1

0

mðrÞ2bryðrÞ2b

dr < 1 : ð5:4Þ

Without loss of generality, we may assume that

ZO

ZO

jDuðxÞ �DuðyÞj2bjx� yjnyðjx� yjÞ2b

dx dy < 1 :

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We now define

lðOÞ ¼ZO

ZO

jDuðxÞ �DuðyÞj2bjx� yjnyðjx� yjÞ2b

dx dy for O � O; O open:

Then l fulfills the hypotheses of Lemma 5.1. Next, we want to apply Proposition 3.1.In order to do this, we have to assume that there exists a constant eAA 2 ð0;1Þ suchthat

yð2rÞ � eAAyðrÞ ð5:5Þ

for every r > 0 (see Proposition 3.1). Then, Proposition 3.1 and Remark 3.2 yield

�Z

Brðx0ÞjDu� ðDuÞx0;rj

2bdx

� cðn; eAA; bÞ yðrÞ2brn

ZBrðx0Þ

ZBrðx0Þ

jDuðxÞ �DuðyÞj2bjx� yjnyðjx� yjÞ2b

dx dy ð5:6Þ

whenever Brðx0Þ � O. Hence, using Holder’s inequality, we see that x0 2 S0 impliesthat

lim supr&0

yðrÞ2brn

lðBrðx0ÞÞ > 0 :

Note that

ð0;1Þ 3 r 7!o1ðrÞ :¼ rn

yðrÞ2b

fulfills the hypotheses (5.2), (5.3). Hence, letting

S0 ¼(y 2 O : lim sup

r&0

yðrÞ2brn

lðBrðyÞÞ > 0

);

we have S0 � S0, and we are in a position to apply Lemma 5.1 to conclude thatHo1ðS0Þ ¼ 0 yielding also that Ho1ðS0Þ ¼ 0. Observe that the modulus yðÞ is stillto be chosen.

Estimate for R1. We will show that S1 is a null set with respect to a sphericalHausdorff measure H~oo for some properly chosen function ~oo. To this end, weconsider

S1 ¼�y 2 O : lim sup

r&0

~ooðrÞ�1lðBrðyÞÞ > 0

�;

1236 Duzaar, Gastel, and Mingione

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and assume that ~oo satisfies the hypotheses of Lemma 5.1, i.e., (5.2) and (5.3). For

x0 2 OnS1, the sequence�~ooðrkÞ�1lðBrkðx0ÞÞ

�k2N0

, rk ¼ 2�kr, is bounded. Using(5.6) we obtain

jðDuÞx0;rkþ1� ðDuÞx0;rk j

2b � 2n �Z

Brk ðx0ÞjDu� ðDuÞx0;rk j

2bdx

� cðnÞyðrkÞ2br�nk l�Brkðx0Þ

�¼ cðnÞsðrkÞ2b ~ooðrkÞ�1l

�Brkðx0Þ

�;

where we have introduced

~ooðrÞ ¼ rn

yðrÞ2bsðrÞ2b ð5:7Þ

with s : ½0;1Þ ! ½0;1Þ to be chosen. Inorder to apply Lemma 5.1, ~ooðrÞ must satisfythe hypotheses (5.2) and (5.3). This can be achieved if we assume that

r 7! yðrÞsðrÞ is non-decreasing with lim

r&0

yðrÞsðrÞ ¼ 0 : ð5:8Þ

Hence

jðDuÞx0;rkþ1� ðDuÞx0;rk j � c sðrkÞ:

In particular, if we impose the Dini conditionZ 1

0

sðrÞr

dr < 1 ;

we see that

jðDuÞx0;rk j � cX1k¼0

sðrkÞ � cðlog 2Þ�1

Z r

0

sðrÞr

dr < 1 :

Having estimated jðDuÞx0;rkj, we can also estimate jðDuÞx0;r j for all r 2 ð0;rÞ; givensuch r, there is a unique k 2 N0 with r 2 ðrkþ1; rk�, and for this k we have

1

2jðDuÞx0;r j

2 � jðDuÞx0;rk j2 þ jðDuÞx0;r � ðDuÞx0;rk j

2

� cþ �Z

Brðx0ÞjDu� ðDuÞx0;rk j

2dx

� cþ cðnÞ�Z

Brk ðx0ÞjDu� ðDuÞx0;rk j

2dx ;

the latter integral being bounded by the argument above. Hence, x0 2 OnS1 impliesx0 2 OnS1, i.e., S1 � S1. Lemma 5.1 then yields H~ooðS1Þ ¼ 0.

Elliptic Systems, Singular Sets and Dini Continuity 1237

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Effective Choice of r and h. From the definition of ~oo, i.e., (5.7), and of o1 wesee that

limr&0

~ooðrÞo1ðrÞ ¼ 0 :

SinceHo1ðS0Þ ¼ 0, Theorem 2.6 implies also thatH~ooðS0Þ ¼ 0. Hence, up to now wehave proved that H~ooðSinguÞ ¼ 0, where ~ooðrÞ is defined in (5.7). It remains to makeeffective choices for sðrÞ and yðrÞ. The following elementary lemma will be useful.

Lemma 5.2. Whenever a modulus of continuity k fulfills the Dini condition

KðrÞ :¼Z r

0

kðrÞr

dr < 1

for some r > 0, then also kK�s satisfies the Dini condition for every s 2 ð�1; 1Þ.

Proof. Since K0ðrÞ ¼ kðrÞr , we have

Z r

0

kðrÞKðrÞ�s

rdr ¼

Z r

0

K0ðrÞKðrÞ�sdr ¼ KðrÞ1�s

1� s< 1 : &

We now come to the proof of Theorem 2.2; we choose, for any e 2 ð0; 1Þ,

yðrÞ :¼ mðrÞgeLLðrÞ 12b� e

4b; sðrÞ :¼ mðrÞgeLLðrÞ e4b�1; eLLðrÞ :¼ Z r

0

mðrÞgr

dr; ð5:9Þ

with a parameter g > 0 to be chosen. By Lemma 5.2 (applied with s ¼ 1� e4b), we

infer that s fulfills the Dini condition provided eLLð1Þ < 1, which means that mðrÞgfulfills a Dini condition. In order to satisfy (5.4), we use Lemma 5.2 (now withs ¼ 1� E

2); we can do so if 2bð1� gÞ ¼ g. This yields:

g :¼ 2b1þ 2b

¼:2

3t1 for some t1 > 1: ð5:10Þ

Note that t1 depends only on n, N , L, n, and d. These choices lead us to

~ooðrÞ ¼ rn�sðrÞyðrÞ

�2b

¼ rneLLðrÞe�3t2 t2 :¼ 1þ 2b :

Moreover, using the concavity of m we see that (5.5) holds with eAA :¼ 2gð1þ1=ð2bÞ�e=ð4bÞÞ.Now, we take

t ¼ tðbÞ :¼ minft1; t2g > 1

and

oðrÞ :¼ rn�Z r

0

m23tðrÞr

dr

�e�3t

¼: rnLðrÞe�3t :

1238 Duzaar, Gastel, and Mingione

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In the case t ¼ t2 < t1, a simple application of L’Hopital’s rule gives

limr&0

oðrÞ~ooðrÞ ¼ 0 ;

providedR 10

mðrÞ23t2r

dr < 1 whereas in the case t ¼ t1 < t2 the same conclusion holds,

providedR 10

mðrÞ23t1r

dr < 1. In the remaining case, we have ~oo ¼ o. SinceH~ooðSing uÞ ¼ 0, an application of Theorem 2.6 yields HoðSing uÞ ¼ 0. This provesthe theorem in the case e 2 ð0; 1Þ. The case e � 1 trivially follows by an application ofTheorem 2.6.

We finally note that t > 1 since b > 1. Moreover, t depends on the sameparameters as the exponent b. This proves Theorem 2.2.

Proof of Theorem 2:1. The proof follows the lines of the proof of Theorem 2.2.From Proposition 4.1, i.e., (4.9), and Proposition 3.3 (with ~mm ¼ mb, q ¼ 2) we see thatDu 2 W

y;2loc ðO;RN Þ for any modulus of continuity yðrÞ which satisfies

Z 1

0

mðrÞ2bryðrÞ2 dr < 1 : ð5:11Þ

Now we can proceed exactly as in the proof of Theorem 2.2; we just have to replace(5.4) by (3.2) and b by 1 and obtain the analogues of (5.6)–(5.8). We note that b < 1has been incorporated in ~mm. Then, we arrive at (5.9) with m, b replaced by mb, 1. Thecorresponding choice of g is g ¼ 2

3, and this implies the assertion of Theorem 2.1 witht :¼ 2

3 b. &

ACKNOWLEDGMENT

The research of Giuseppe Mingione is partially supported by MIUR via theproject ‘‘Calcolo delle Variazioni’’ (PIN 2002).

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Received July 2003Accepted January 2004

1240 Duzaar, Gastel, and Mingione

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