Boundary Regularity in Variational Problems

Report no. OxPDE-08/08

Oxford Centre for Nonlinear PDE Mathematical Institute, University of Oxford Radcliffe Observatory Quarter, Gibson Building Annexe Woodstock Road Oxford, England OX2 6HA Email: [email protected] September, 2008

Boundary regularity in variational problems

By Jan Kristensen

Mathematical Institute - University of Oxford &

Giuseppe Mingione Università degli Studi di Parma

BOUNDARY REGULARITY IN VARIATIONAL PROBLEMS

JAN KRISTENSEN AND GIUSEPPE MINGIONE

Abstract. We prove that, if u : Ω → RN is a solution to the Dirichlet varia-tional problem

minw

Z

ΩF (x, w, Dw) dx subject to w ≡ u0 on ∂Ω ,

with regular boundary datum (u0, ∂Ω) and regular integrand F (x, w, Dw),then almost every boundary point, in the sense of the usual surface measureof ∂Ω, is regular for u in the sense that Du is Holder continuous in a relativeneighborhood of the point. We remark that the question of existence of evenone such regular boundary point was open except for very special cases. Theresult is a consequence of a new global higher differentiability result that weestablish for minima of the functionals in question. Our methods also allow usto improve the known boundary regularity results for solutions to non-linearelliptic systems, and in some cases, to improve the known interior singular setsestimates for minimizers.

Contents

1. Introduction and results 12. Preliminaries 113. Fractional Sobolev spaces 144. Basic regularity results 165. Comparison lemmas 216. Conditional fractional estimates 347. Existence of regular boundary points 428. Variational Calderon-Zygmund estimates 47References 61

1. Introduction and results

The problems in question are of the type

(1.1) minwF(w)

where

(1.2) F [w] :=∫

Ω

F (x,w,Dw) dx ,

and the minimization is over W 1,p Sobolev mappings w : Ω ⊂ Rn → RN satisfyinga Dirichlet boundary condition w = u0 on ∂Ω. Furthermore we assume for anα ∈ (0, 1] that

(1.3)

Ω is an open bounded C1,α domain in Rn

u0 ∈ C1,α(Ω,RN ) .

Version: August 13, 2008.

1

2 JAN KRISTENSEN AND GIUSEPPE MINGIONE

We are concerned with the multi-dimensional vectorial case n, N ≥ 2 and considerproblems with quadratic/superquadratic growth at infinity p ≥ 2.

Under suitable convexity and smoothness assumptions related to p on F (·) - seenext section for the precise assumptions - it turns out that solutions u ∈ u0 +W 1,p

0 (Ω,RN ) to (1.1) are partially regular: the gradient Du is Holder continuousoutside a negligible relatively closed subset of Ω. It is natural to inquire if thisregularity result has an extension to the boundary ∂Ω of Ω. The only knownanswer is basically in the work of Jost & Meier [30] who considered variants of thespecial functional

(1.4) w 7→∫

Ω

c(x,w)|Dw|2 dx ,

where c(·) is a Holder continuous and bounded function which is uniformly boundedaway from zero. They proved that minimizers are Holder continuous up to theboundary but did not address the question of higher order regularity. The mainfeature of the functional in (1.4) is that its energy density depends on the gradientvariable only via its euclidean norm |Dw| - this renders the situation very specialand allows the authors to use powerful techniques that are not available for moregeneral functionals. In fact full boundary regularity follows in such cases, a resultthat is known to fail in the general vectorial case (see [23]). These special techniquesare in turn related to the fundamental result of Uhlenbeck [52], and indeed theabove special energy density is sometimes said to have “Uhlenbeck structure” - seefor instance [8]. Apart from such very special structures, in the general case offunctionals, including for instance

(1.5) w 7→∫

Ω

c(x,w)f(Dw) dx and w 7→∫

Ω

c(x)f(Dw) + h(x,w) dx ,

the problem of boundary regularity of minima has remained essentially untouched,and the criteria for boundary regularity available in the literature do not yieldthe existence of even one regular boundary point, neither for the gradient of thesolution Du, nor for u itself. This is a completely unsatisfying situation, especiallyif compared to the scalar case - i.e. real-valued minima - where solutions are againknown to be everywhere regular up to the boundary.

The aim of this paper is to initiate the theory of partial regularity at the bound-ary for general variational problems of Dirichlet type. One of our main resultsstates that for a large class of functionals, including in particular (1.5), almost ev-ery boundary point in the sense of the usual surface measure is a regular point for asolution u, in the sense that Du is Holder continuous in a relative neighborhood ofsuch points. We prove, in other words, that partial regularity of minima naturallyextends up to the boundary.

Our approach is to first prove global higher differentiability of a minimizer u.With the right choice of assumptions on F this global higher differentiability inturn implies that Du admits a trace on the boundary ∂Ω, and in combinationwith a criterion for regularity of points at the boundary the up-to-the-boundarypartial regularity follows. In fact, we shall give several results of such type, andthe existence of regular boundary points will be given along with estimates on theHausdorff dimension of the boundary singular set of minima. In the process wedevelop techniques that also allow us to improve the boundary regularity resultsfor elliptic systems obtained in [17], and in some cases the interior singular setsestimate obtained in [36] (see Remark 7.2 below).

1.1. Description of the problem and first results. In order to describe ourviewpoint and state the main results, let us start by recalling the boundary regu-larity results for solutions to non-linear elliptic systems recently obtained in [17].

https://www.researchgate.net/publication/247379119_Regularity_Results_for_Nonlinear_Elliptic_Systems_and_Applications?el=1_x_8&enrichId=rgreq-73e21a04-6bca-4c9a-acbe-42c13ccdc2a2&enrichSource=Y292ZXJQYWdlOzIyNTE1NzYwNztBUzoxMDE4OTcxMTc2MzQ1NjVAMTQwMTMwNTU3NDU2OA==

https://www.researchgate.net/publication/265330333_Multiply_Integrals_in_the_Calculus_of_Variations_and_Nonlinear_Elliptic_Systems_Ann_Math_Stud?el=1_x_8&enrichId=rgreq-73e21a04-6bca-4c9a-acbe-42c13ccdc2a2&enrichSource=Y292ZXJQYWdlOzIyNTE1NzYwNztBUzoxMDE4OTcxMTc2MzQ1NjVAMTQwMTMwNTU3NDU2OA==

https://www.researchgate.net/publication/226369006_Boundary_regularity_for_minima_of_certain_quadratic_functionals?el=1_x_8&enrichId=rgreq-73e21a04-6bca-4c9a-acbe-42c13ccdc2a2&enrichSource=Y292ZXJQYWdlOzIyNTE1NzYwNztBUzoxMDE4OTcxMTc2MzQ1NjVAMTQwMTMwNTU3NDU2OA==

https://www.researchgate.net/publication/225747930_Uhlenbeck_K_Regularity_for_a_class_of_non-linear_elliptic_systems_Acta_Math_138_219-240?el=1_x_8&enrichId=rgreq-73e21a04-6bca-4c9a-acbe-42c13ccdc2a2&enrichSource=Y292ZXJQYWdlOzIyNTE1NzYwNztBUzoxMDE4OTcxMTc2MzQ1NjVAMTQwMTMwNTU3NDU2OA==

BOUNDARY REGULARITY IN VARIATIONAL PROBLEMS 3

We consider the following Dirichlet problem involving a non-linear homogeneouselliptic system:

(1.6)

div a(x, u,Du) = 0 in Ωu = u0 on ∂Ω.

Here the vector field a : Ω × RN × RNn → RNn is continuous and satisfies thefollowing standard growth, ellipticity and continuity assumptions:

(1.7)

|a(x, y, z)|+ |az(x, y, z)|(1 + |z|2) 12 ≤ L(1 + |z|2) p−1

2

ν(1 + |z|2) p−22 |λ|2 ≤ 〈az(x, y, z)λ, λ〉

|a(x1, y1, z)− a(x2, y2, z)| ≤ Lωα(|x1 − x2|+ |y1 − y2|)(1 + |z|2) p−12 ,

for any z, λ ∈ RNn, y, y1, y2 ∈ RN and x, x1, x2 ∈ Ω, where p ≥ 2, n,N ≥ 2,0 < ν ≤ L, and the dependence on the coefficients (x, y) is Holder continuous withexponent α

(1.8) ωα(s) := minsα, 1 α ∈ (0, 1] .

We recall that the set of regular points defined by

(1.9) Ωu := x ∈ Ω : Du is continuous on some neighbourhood A of x ,by definition is open, and it turns out that |Ω\Ωu| = 0. Its complement, denoted byΣu := Ω\Ωu, is called the singular set of u. As a matter of fact, under the previousassumptions Du ∈ C0,α

loc (Ωu,RNn). Similar definitions and partial regularity resultsalso hold in the case of minimizers. In this case one has Du ∈ C0,α/2

loc (Ωu,RNn).By now standard references on partial regularity include [25, 23, 20, 2, 18, 31, 45,46, 50, 51]; an up-dated presentation is in [42].

When dealing with Dirichlet problems as (1.1) and (1.6) it is then natural todeal with the set of regular boundary points defined as

(1.10) Ωbu := x ∈ ∂Ω : u ∈ C1(Ω ∩A,RN ) for some neighborhood A of x .

The results obtained in [17] state that under the assumption

(1.11) α >12

in the general case of homogeneous systems of the type

(1.12) div a(x,Du) = 0 for every n ≥ 2 ,

and for the general homogeneous systems in the low dimensional case

(1.13) div a(x, u,Du) = 0 when n ≤ p+ 2 ,

Hn−1-almost every boundary point is regular in the above sense. The strategy of[17] starts from the interior singular sets dimension estimates in [40, 41] assertingthat in both cases (1.12)-(1.13) it is dimH(Σu) ≤ n− 2α. Then, extending such anestimate up to the boundary and using (1.11) gives

(1.14) dimH(∂Ω \ Ωbu) ≤ n− 2α < n− 1 = dimH(∂Ω) ,

and therefore the almost everywhere - in the sense of the Hn−1 surface measure -boundary regularity follows. The assumption (1.11) has also another natural mean-ing: it guarantees that Du has a trace Du|∂Ω on ∂Ω. The above mentioned partialregularity result in [17] is derived as a consequence of a higher differentiabilityresult, namely that

(1.15) Du ∈W 2α−εp ,p(Ω,RNn) , for every ε > 0.

The trace theorem (see [4]) then yields

Du|∂Ω ∈W2α−1−ε

p ,p(∂Ω,RNn) ,


provided (1.11) holds.Returning to the case of minimizers we remark that the general class of func-

tionals considered here with merely Holder continuous coefficients as in (1.5) do notadmit an Euler-Lagrange system, and consequently the regularity theory availablefor systems does not apply to minimizers. However even when the functional doesadmit an Euler-Lagrange system it is well-known that it might not be possible toestablish partial regularity from the Euler-Lagrange system alone. It is importantin these situations that we are dealing with a minimizer and not merely a solutionto the Euler-Lagrange system, and for this alternative methods are needed.

In a first attempt to obtain for minimizers results similar to those available forthe cases (1.12)-(1.13), the idea of extending the available singular sets estimatesobtained in [36] up to the boundary cannot work. In fact, even in the most favorablecases the interior singular set results available for functionals would give an estimateof the type dimH(Σu) < n − α, which is clearly insufficient for the argument in(1.14). Therefore in this paper we shall first develop a technique for improving theavailable singular sets estimates for minima up to the ones available for systems,hereby obtaining dimH(Σu) < n − 2α, or the like, in the interior, and then weshall extend this last inequality up the boundary, eventually using the argument in(1.14). Both steps require new and delicate arguments. This will lead us to obtainfor minimizers a theory comparable to the one developed in [17] for systems. Itapplies to a large class of functionals that includes most of the usual model casesconsidered in the literature, and in particular those in (1.5).

Let us start from a first but nevertheless significant model result, concerning themodel functional

(1.16) F1[w] :=∫

Ω

c(x)f(Dw) + d(w)(1 + |Dw|2) γ2 dx .

We shall assume the following standard Holder continuity of the coefficients:

(1.17) 0 < ν ≤ c(x) ≤ L , c(·) ∈ C0,α(Ω) ,

whenever x ∈ Ω, and

(1.18) 0 ≤ d(y) ≤ L , d(·) ∈ C0,β(Ω) ,

whenever y ∈ RN , while, as usual, the following strong convexity of f(·) is assumedto hold whenever z, λ ∈ RNn:

(1.19) ν|z|p ≤ f(z) , ν(1 + |z|2) p−22 |λ|2 ≤ 〈fzz(z)λ, λ〉 ≤ L(1 + |z|2) p−2

2 |λ|2 .Theorem 1.1. Under the assumptions (1.17)-(1.19) with

(1.20) α >12

β >23

γ ≤ 2p3,

let u ∈ u0 +W 1,p0 (Ω,RN ) be a solution to the Dirichlet problem (1.1) with (1.3) and

F [·] ≡ F1[·]. Then Hn−1-almost every boundary point is regular for u. Moreover,the global higher differentiability

(1.21) Du ∈W 1p +ε,p(Ω,RNn) and Du|∂Ω ∈W ε,p(∂Ω,RNn) ,

hold for some ε ≡ ε(α, β) > 0.

The previous theorem, that can be considered as a first analog of the resultvalid for (1.12) and that we reported as a first case for expository purposes, willbe largely improved later, and it is actually a particular case of results applying tomuch more general functionals including those of the type

(1.22) F2[w] :=∫

Ω

f(Dw) + h(x,w,Dw) dx ,


where an essential feature for our approach is that the function h(·) grows at asuitably lower rate

(1.23) 0 ≤ h(x, y, z) ≤ L(1 + |z|2)γ/2 γ < p .

We refer to Theorem 1.9 below for the precise statement, and to Section 3 for thedefinition of the space in (1.21). We also refer to (7.19) for a statement alternativeto (1.21), where we recall that the second inclusion follows by the first.

Remark 1.1. The assumption (1.23) is suggested by the role played by the coeffi-cients in the functional. We have seen that in order to obtain estimates for singularsets we need to bound the oscillations with respect to the x variable, and this isthe meaning of (1.11). When considering an energy density depending on u we canre-write F (x, u(x), Du) ≡ F (x,Du), and in this sense u acts as a measurable coef-ficient making the significance of (1.11) less clear. The role of an assumption like(1.23) emerges at this stage: the perturbation due to the oscillations of u(x) canaffect the leading regularizing term only at the lower rate γ. If γ is small enoughcompared to p, then it turns out that we can efficiently estimate the singular set.For the same reason, we have to assume a bit more regularity on the partial functiony 7→ F (·, y, ·) than assumed on x 7→ F (x, ·, ·). Anyway, we stress that the centralrole here is played by α, which is linked to the explicit presence of x.

We next turn to the low dimensional case, that is when n ≤ p+2. We remark thatboth in the case of systems [17, 41] and for functionals [36], the effect of n ≤ p+ 2is to, by a suitable Sobolev embedding, obtain a better control of the oscillations ofthe map u(x) and hence on the coefficients x 7→ a(x, u(x), ·) and x 7→ F (x, u(x), ·),respectively. We shall consider the general assumptions(1.24)

ν|z|p ≤ F (x, y, z) ≤ L(1 + |z|2) p2

ν(1 + |z|2) p−22 |λ|2 ≤ 〈Fzz(x, y, z)λ, λ〉 ≤ L(1 + |z|2) p−2

2 |λ|2|F (x1, y1, z)− F (x2, y2, z)| ≤ L [ωα(|x1 − x2|) + ωβ(|y1 − y2|)] (1 + |z|2) p

2

|Fz(x1, y1, z)− Fz(x2, y2, z)| ≤ Lωα(|x1 − x2|+ |y1 − y2|)(1 + |z|2) p−12 ,

satisfied for all x, x1, x2 ∈ Ω, y, y1, y2 ∈ RN and z, λ ∈ RNn, where p ≥ 2,0 < ν ≤ L, and the dependence on the coefficients x and y is Holder continuouswith exponents α and β, respectively. Thus

(1.25) ωα(s) := minsα, 1 and ωβ(s) := minsβ , 1 0 < α ≤ β ≤ 1.

Except for (1.24)4, these assumptions are standard in the context of criteria forpartial interior regularity and for boundary regularity. Condition (1.24)4 is lesscommon and sometimes used to obtain sharper integrability results [24]. Note thatit is anyway mild and automatically satisfied in many cases, including those in (1.5),that in turn give a very large part of the examples in the literature.

The result is now the following theorem, which can be considered as the naturalcounterpart of the boundary regularity result in (1.13):

Theorem 1.2. Under the assumptions (1.24) with

(1.26) α >12

β > max

1− 2n,23

n ≤ p+ 2 ,

let u ∈ u0 + W 1,p0 (Ω,RN ) be a solution to the Dirichlet problem (1.1) with (1.3).

Then Hn−1-almost every boundary point is regular for u and (1.21) holds.

Remark 1.2. For the rest of the paper, we confine ourselves to the case

(1.27) n ≥ 3 ,


since when n = 2 solutions are everywhere continuous up to the boundary - seeSection 4.1 below.

1.2. Relaxing the Holder continuity of coefficients. Here we state the mainresults of the paper, Theorems 1.3-1.5 below. Our concern is to relax the assumption(1.11), allowing the map x 7→ Fz(x, ·, ·) to be Holder continuous with an arbitrarilysmall exponent σ > 0. More precisely the viewpoint is the following: since theHolder continuity of the coefficient is used to prove that the gradient lies in asuitable fractional Sobolev space as in (1.15), which in turn implies the singularsets estimates, then it seems plausible that the same result should hold assumingthat the coefficients of the functional are themselves in a fractional Sobolev space.Therefore we shall compensate a very mild Holder continuity condition on thecoefficient by a fractional Sobolev type differentiability assumption on the mapx 7→ Fz(x, ·, ·). The results in the previous section will appear as particular casesof the ones we are proposing here.

Let us start with the rough coefficients counterpart of Theorem 1.1, where theassumptions on the coefficient c(·) are considerably weakened. We shall replace theHolder dependence on x by the following:

(1.28)

0 < ν ≤ c(x) ≤ L , c(·) ∈Wα,ns(Ω) ∩ C0,σ(Ω) for some σ > 0

ns < n n− ns depends on n,N, p, L/ν

and therefore assuming Holder continuity with exponent α at the lower Lns inte-grability scale, and standard Holder regularity - L∞ scale - only with a potentiallysmall exponent σ. See Section 4.2 and (4.8) for the precise choice of the numberns.

In order to formulate assumptions similar to (1.28) for more general functionalsof the types (1.2) and (1.22), where x-coefficients do not split as in (1.16), we haveto look for an alternative way of stating fractional differentiability. A useful tool isprovided by the work of DeVore & Sharpley [14], who noticed that for 0 < s < 1and 1 < q <∞, if c(·) ∈W s,q(Rn) then there exists g ∈ Lq(Rn) - one may take thes-fractional sharp maximal function of c(·) - such that, for almost all x1, x2 ∈ Rn

it holds that

(1.29) |c(x1)− c(x2)| ≤ (g(x1) + g(x2))|x1 − x2|s .We remark that in the case s = 1, so that c(·) ∈ W 1,q, one can take g ≈ M(|Dc|);see also [1] and [9]. The authors of [14] thereby define a new function space Cs

q ,saying that

c(·) ∈ Csq (Rn)⇐⇒ (1.29) holds for some g ∈ Lq(Rn) .

These spaces are generally not Besov spaces, but are nevertheless comparable tothem and to the usual fractional Sobolev spaces in the sense that Wα,q(Rn) ⊂Cα

q (Rn) ⊂ Wα−ε,q(Rn), for every ε ∈ (0, α). Definition (1.29) provides us withthe right setting and we shall work with functionals with coefficients in Cα

ns; note

indeed that Cαns⊃ Wα,ns ⊃ Wα,∞ ≡ C0,α on bounded domains. We shall use the

following set of assumptions:

(1.30)

ν|z|p ≤ F (x, y, z) ≤ L(1 + |z|2) p2

ν(1 + |z|2) p−22 |λ|2 ≤ 〈Fzz(x, y, z)λ, λ〉 ≤ L(1 + |z|2) p−2

2 |λ|2|F (x, y1, z)− F (x, y2, z)| ≤ Lωβ(|y1 − y2|)(1 + |z|2) γ

2

|Fz(x1, y, z)− Fz(x2, y, z)| ≤ (g(x1) + g(x2))|x1 − x2|α(1 + |z|2) p−12

|Fz(x, y1, z)− Fz(x, y2, z)| ≤ Lωβ(|y1 − y2|)(1 + |z|2) γ−12


where, (1.30)4,5 replace (1.24)4, ωα(·), ωβ(·) are as in (1.25), and the rest is as in(1.24) with

(1.31) γ ≤ p and 0 ≤ g ∈ Lns(Ω) , ns < n ,

where ns is as in (1.28) and (4.8) below. The function g(·) plays in other words therole of an α-th derivative of the function x 7→ Fz(x, ·, ·) and so (1.30)4 describes theα-Holder continuity at a weaker rate, since the “Holder constant g(·)” may blow-upat some points.

Of course we shall also assume that

(1.32)

|F (x1, y1, z)− F (x2, y2, z)| ≤ Lωσ(|x1 − x2|+ |y1 − y2|)(1 + |z|2) p

2

ωσ(s) = minsσ, 1 , for some σ > 0 .

This is essential for any known proof of partial regularity - without which therewould be no singular set to estimate; the point is that we are no longer requiringσ > 1/2. As a matter of fact, with the notation of (1.9)-(1.10), we have under thehypotheses (1.30)1,2 and (1.32) that Du ∈ C0,σ/2

loc (Ωu ∪ Ωbu,RNn).

We start by stating the results in the low dimensional case. In the next theoremwe have no restriction on γ except for the natural γ ≤ p.Theorem 1.3. Under the assumptions (1.30)-(1.32), assume that

(1.33) n ≤ p+ 2 α >12

and β > max

1− 2n,23

.

Let u ∈ u0 +W 1,p0 (Ω,RN ) be a solution to the Dirichlet problem (1.1) with (1.3).


The assumption n ≤ p+ 2 can also be slightly weakened; see Remark 7.1 below.There is an intermediate low dimensional case worth reporting on here.

Theorem 1.4. Under the assumptions (1.30)-(1.32) with γ < p, assume that

(1.34) n ≤ 2p+ 2 α >12

and β > max

1− 2n,23

.



Note that when increasing the dimension bound from (1.33) to (1.34) we com-pensate by assuming γ < p. This phenomenon is amplified when passing to thegeneral high dimensional case, as also suggested by Remark 1.1.

Theorem 1.5. Under the assumptions (1.30)-(1.32) with γ < p, take

(1.35)23≤ s ≤ p

p+ 1and assume that

(1.36) α >12

β > s γ ≤ ps+2psn− 2

.



The previous statement should be understood as follows: (α, β, γ) characterizethe structure of the integrand F (·) via (1.30), while the parameter s, when varyingin the range (1.35), can be considered to parametrize the various results. We notethat (1.36) is in perfect accordance with the principle suggested in Remark 1.1.Condition (1.36) states that the less regularity we assume on y 7→ F (·, y, ·), the


less it can be allowed to grow with respect to y. The extreme cases of assumptions(1.35)-(1.36) are given by s = 2/3 corresponding to

α >12

β >23

γ ≤ 2p3

+4p

3(n− 2),

and by s = p/(p+ 1) corresponding to

α >12

β >p

p+ 1γ ≤ p2

p+ 1+

2p2

(n− 2)(p+ 1).

Thus strong growth of y 7→ F (·, y, ·) and y 7→ Fz(·, y, ·) is compensated by a higherdegree of Holder continuity of y 7→ F (·, y, ·) and y 7→ Fz(·, y, ·).

Note also that Theorems 1.1 and 1.2 correspond to particular cases where g(·) ∈L∞ in Theorems 1.5 and 1.3, respectively.

Remark 1.3. We remark that assumptions (1.30) in particular cover the case(1.16) of splitting coefficients.

Remark 1.4. The approach in this section, when applied to non-linear ellipticsystems (see Section 1.4) allows us to answer in the negative the question posedin [17] as to whether or not the C0,α-Holder continuity of (x, y) 7→ a(x, y, ·) withα > 1/2 is really needed to prove boundary partial regularity.

Remark 1.5. The appearance of the space Wα,ns with n > ns in (1.28) andin (1.31), somehow resembles the role played for integrability of solutions to non-divergence form equations of the type

(1.37) F (x,D2u) = h(x) ,

see also [19]. Indeed, roughly speaking, after differentiating - in some potentiallyfractional sense - equations such as div c(x)a(Du) = 0, we come to something similarto (1.37), where the fractional derivative of c(·) plays the role of h(·) in (1.37). Forearlier results involving equations with Sobolev coefficients see [12, 22, 44].

1.3. Relevant model cases. Here we list a few particular cases of the Theorems1.3-1.5 which deserve a separate statement; these involve relevant model cases usu-ally treated in the literature. Moreover, we treat an additional case not covered bythe general theorems presented above. We consider functionals of the form (1.16)with a more general integrand f : RN × RNn → R as in (1.24)

(1.38)

ν|z|p ≤ f(y, z) ≤ L(1 + |z|2) p2

ν(1 + |z|2) p−22 |λ|2 ≤ 〈fzz(y, z)λ, λ〉 ≤ L(1 + |z|2) p−2

2 |λ|2|f(y1, z)− f(y2, z)| ≤ Lωβ(|y1 − y2|)(1 + |z|2) γ

2

|fz(y1, z)− fz(y2, z)| ≤ Lωβ(|y1 − y2|)(1 + |z|2) γ−12 .

Theorem 1.6. Let u ∈ u0 + W 1,p0 (Ω,RN ) be a solution to the Dirichlet problem

(1.1) with (1.3) and

(1.39) F [w] =∫

Ω

c(x)f(w,Dw) dx .

Assume that the function f(·) satisfies (1.38), that c(·) satisfies (1.28), and finallyassume (1.36) with γ < p. Then Hn−1-almost every boundary point is regular foru and (1.21) holds. The same holds replacing (1.36) by (1.34), or replacing (1.36)by (1.26) with γ = p.



(1.1) with (1.3) and

(1.40) F [w] =∫

Ω

c(x)d(w)f(Dw) dx .

Assume that f(·) satisfies (1.19), that c(·) satisfies (1.28) with α > 1/2, and thatd(·) satisfies (1.18) with β > max1 − 2/n, 2/3. If n ≤ p + 2, then Hn−1-almostevery boundary point is regular for u and (1.21) holds.

For certain special energies, we may allow even a discontinuous dependence ofthe integrand on the variable x; according to the explanations of Remark 1.1 thishappens when x does not affect the regularizing part of the integrand, i.e. the onecontaining Du. We consider an integrand f : Ω×RNn → R satisfying (1.30)-(1.32),obviously recast for the case with no dependence on u(x):

(1.41)

ν|z|p ≤ f(x, z) ≤ L(1 + |z|2) p2

ν(1 + |z|2) p−22 |λ|2 ≤ 〈fzz(x, z)λ, λ〉 ≤ L(1 + |z|2) p−2

2 |λ|2

|fz(x1, z)− fz(x2, z)| ≤ (g(x1) + g(x2))|x1 − x2|α(1 + |z|2) p−12 .

Here g(·) is as in (1.31). Moreover we shall consider another Caratheodory functionh : Ω× RN → [0,∞) such that

(1.42)

|h(y)| ≤ L(1 + |y|)γ

|h(x, y1)− h(x, y2)| ≤ Lωβ(|y1 − y2|)(1 + |y1|+ |y2|)γγ < p

for every y, y1, y2 ∈ RN , where ωβ(·) is as in (1.25).


(1.1) with (1.3) and

(1.43) F [w] =∫

Ω

f(x,Dw) + h(x,w) dx .

Assume that the function f(·) satisfies (1.41) and (1.32), that h(·) satisfies (1.42),and assume (1.36). Then Hn−1-almost every boundary point is regular for u and(1.21) holds.

The significance of the previous result lies in the fact that the function x 7→ h(x, ·)is a priori only measurable. It is indeed somehow surprising to obtain regularboundary points although having arbitrarily discontinuous coefficients. A typicalmodel example in this case is given by

w 7→∫

Ω

c(x)f(Dw) + c1(x)|u|γ dx ,

where c1(·) is a bounded non-negative measurable function and γ ≥ β. Finallythe result for functionals of the type in (1.22), implied by Theorem 1.5. We shallassume that


(1.1) with (1.3), and F [·] ≡ F2[·] defined in (1.22). Here the function f(·) satisfies(1.41) and the function h(·) satisfies (1.30), when obviously recast for the case withno x-dependence. If (1.36) holds, then Hn−1-almost every boundary point is regularfor u and (1.21) holds.


1.4. New results for elliptic systems. As mentioned in Remark 1.4, the ap-proach of the previous section allows us to improve the results in [17] in thatwe can find partial boundary regularity assuming Holder continuity of coefficientswith an arbitrarily small exponent σ > 0. We aim at having the results availablein (1.12)-(1.13) when the x-coefficients exhibit fractional Sobolev dependence. Weshall moreover consider more general cases than those of [17]. The assumptions onthe Caratheodory vector field a : Ω× RN × RNn → RNn are now

(1.44)

|a(x, y, z)|+ |az(x, y, z)|(1 + |z|2) 12 ≤ L(1 + |z|2) p−1

2

ν(1 + |z|2) p−22 |λ|2 ≤ 〈az(x, y, z)λ, λ〉

|a(x1, y, z)− a(x2, y, z)| ≤ (g(x1) + g(x2))|x1 − x2|α(1 + |z|2) p−12

|a(x1, u, z)− a(x2, y, z)| ≤ Lωα(|y1 − y2|)(1 + |z|2) γ−12 ,

with the same meaning as in (1.7), but now γ ≤ p and g(·) ∈ Lns(Ω) as in (1.31);obviously ωα(·) is as in (1.8). In order to have partial regularity and therefore asingular set to estimate, we shall again assume that

(1.45)

|a(x1, y1, z)− a(x2, y2, z)| ≤ Lωσ(|x1 − x2|+ |y1 − y2|)(1 + |z|2) p−1

2

ωσ(s) = minsσ, 1 , for some σ > 0 .

A model case for the previous assumptions is given by the system

(1.46) div (c(x)a(Du) + b(u,Du)) = 0

where a(·) satisfies (1.44)1,2, c(·) satisfies (1.28) and finally b(·) is a differentiableand monotone vector field satisfying (1.44)4.

The first result we obtain extends the one in [17] for the case (1.12):

Theorem 1.10. Under the assumptions (1.44)-(1.45) with γ < p, assume that

(1.47) α >12

γ ≤ p− 12

+p

n− 2,

and let u ∈ u0+W 1,p0 (Ω,RN ) be a solution to the Dirichlet problem (1.6) with (1.3).


Recall (1.27). Again in the low dimensional case n ≤ p+ 2 we have

Theorem 1.11. Under the assumptions (1.44)-(1.45) with

α >12

n ≤ p+ 2 ,

let u ∈ u0 + W 1,p0 (Ω,RN ) be a solution to the Dirichlet problem (1.6) with (1.3).


Remark 1.6. The methods and the results of this paper form the starting pointfor further developments including functionals and systems with more irregularcoefficients and singular sets involving a lower type of regularity - we refer theinterested reader to the forthcoming paper [38].

1.5. Plan of the paper - Technical aspects. After a brief discussion of nota-tion and some preliminary results in Section 2, and clarifying the notions neededconcerning fractional Sobolev spaces in Section 3, in Section 4 we shall give a fewpreliminary regularity results about basic non-linear elliptic systems. These areof two types. The first are stated in terms of Morrey spaces - see Theorems 4.1and 4.2 - and are up-to-the-boundary versions of results due to Campanato [11].These results are crucial in the proofs of our results involving the low dimensionalassumption n ≤ p + 2, as for instance Theorem 1.3. The second type of results,


contained in Section 4.3, are proper manipulations of certain boundary Cacciop-poli estimates valid for general systems. Here we have to be very careful since theprecise form of these estimates play a decisive role in the subsequent estimates. InSection 5 we develop a few comparison lemmas between the solution to (1.1), andsolutions to certain properly regularized problems. Two points are important here:the use of certain Calderon-Zygmund type estimates on small domains - eventuallyproved in Section 8 - and that the original solution is assumed to be in a betterSobolev space, i.e. u ∈ W 1,q for some q > p. This additional assumption will bealways satisfied by virtue of the results in Section 8. In Section 6 we prove the cru-cial fractional gradient differentiability estimates under the additional integrabilityassumption u ∈ W 1,q; here the starting point is a variational difference quotientstechnique introduced in [34, 36] that will be upgraded via a delicate mollificationprocedure, and the comparison estimates of Section 5. Moreover, thanks to theuniform Calderon-Zygmund type estimates on small domains from Section 5, andvia a covering argument, this technique will be shown to work at the boundary too.In Section 7 we shall prove the main results of the paper using as a starting pointthe estimates of Section 6: here the additional integrability assumption u ∈ W 1,q

will be removed combining the higher integrability results of Section 8, and thepartial boundary regularity results of Section 4. In Section 8 we prove the higherintegrability results needed in the preceding sections. Here we exploit in a moresubtle way the interaction, already identified in [40, 41, 34, 36], between the higherintegrability of the gradient and singular sets estimates. In turn, such an interac-tion is partially based on a lower level interaction between certain basic Morreyspace regularity estimates for the gradient and its higher integrability. We shall fi-nally see that the assumptions considered and already crucial for getting fractionaldifferentiability estimates will be the same that are responsible for the higher inte-grability exponents needed as a starting integrability assumption for the fractionalestimates of Section 6. We mention that the use of Calderon-Zygmund estimates toprove singular sets estimates has been introduced in [36], but here their use is verydifferent: while in [36] these were used once to implement the localization technique- they were in fact used on small balls - here they are used twice: first at a globallevel and interacting with preliminary Morrey type regularity, and then at a locallevel, i.e. on properly small domains, when performing certain boundary estimates.For this reason we shall need several types of such estimates, and, in contrast to[36], these estimates must be proved directly for minimizers - of functionals whichin general do not admit an Euler-Lagrange system - and will require a differentapproach.

Acknowledgment. The authors are supported by the ERC grant 207573 “Vec-torial Problems”. The second-named author also acknowledge the hospitality andsupport from the OxPDE Centre at the Mathematical Institute - Oxford, in August2008.

2. Preliminaries

2.1. Basic notation. In this paper we follow the usual convention and denote byc a general constant larger than one, that may vary from line to line. Relevantdependence on parameters and special constants will be properly emphasized inparentheses or by subscripts (e.g. c1, c∗), respectively. All the norms we use on Rn,RN and RnN etc will be standard euclidean ones and denoted by | · | in all cases.With x0 ≡ ((x0)1, . . . , (x0)n) ∈ Rn, we denote

BR(x0) ≡ B(x0, R) := x ∈ Rn : |x− x0| < R


andQR(x0) ≡ Q(x0, R) := x ∈ Rn : sup

i|xi − (x0)i| < R ,

the open ball and cube, respectively, with center x0 and radius/half-sidelength R. Inthe sequel we shall refer to R as a radius also when dealing with cubes Q(x0, R). Inthe rest of the paper all the cubes considered will have sides parallel to the coordinateaxes, that is, they are balls in the `∞ metric on Rn. We shall often use the shorthand notation BR ≡ B(0, R) and QR ≡ Q(0, R) or even BR ≡ B(x0, R) andQR ≡ Q(x0, R) when the center will not be relevant in the context. We also denote

ΓR := QR ∩ x ∈ Rn : xn = 0 = (xi)i≤n : |xi| ≤ R, xn = 0 .Finally we set

(2.1)

B+(x0, R) := B(x0, R) ∩ x ∈ Rn : xn > 0

B+R := BR ∩ x ∈ Rn : xn > 0 ,

and

(2.2)

Q+(x0, R) := Q(x0, R) ∩ x ∈ Rn : xn > 0

Q+R := QR ∩ x ∈ Rn : xn > 0 .

The sets in (2.1)-(2.2) will be called “upper cubes” and “upper balls”whenever x0 ∈ xn = 0; in the last case these are actually “half-cubes” or“rectangles”, i.e. sets of the type

(2.3)

R(x0, R) := Q(x0, R) ∩ x ∈ Rn : xn > (x0)n

(x0) = ((x0)1, . . . , (x0)n−1, (x0)n +R/2) .

Similarly, the sets in (2.1) will be called upper balls or half-balls whenever x0 ∈xn = 0. Accordingly we define

B− := B \B+ and Q− := Q \Q+

and the like. Finally, when Q is a cube, or a rectangle or a ball, we denote by λQthe corresponding cube, rectangle or ball obtained by dilating Q about its center bythe factor λ > 0. We call the attention of the reader to the fact that since we havedefined the upper cubes as those half-cubes with the lower side lying on xn = 0then if Q is an upper cube then λQ is not an upper cube when λ 6= 1.

If h : A→ Rk is an integrable map and 0 < |A| <∞ we write

(h)A := −∫

A

h(x) dx :=1|A|

∫

A

h(x) dx .

Finally, throughout the paper es1≤s≤n denotes the standard basis of Rn. Moreover, to emphasize the role of the last coordinate, we shall write x ≡ (xi)i≤n ≡(x′, xn) ∈ Rn−1 × R.

When considering a function space X(Ω,Rk) of possibly vector valued measur-able maps defined on an open set Ω ⊂ Rn, with k ∈ N, e.g.: Lp(Ω,Rk),W β,p(Ω,Rk),we define the local variant Xloc(Ω,Rk) as that space of maps h : Ω→ Rk for whichh|Ω′ ∈ X(Ω′,Rk), for every Ω′ b Ω. Moreover, also in the case h is vector valued,that is k > 1, we shall also use the short-hand notation X(Ω,Rk) ≡ X(Ω), or evenX(Ω,Rk) ≡ X when the domain and range are clear from the context.

For the rest of the paper, with t ∈ (0, 1] we denote, according to the notationestablished in the Introduction,

ωt(s) := minst, 1 s ≥ 0 .

With c ≥ 1 and s, s1, s2 ≥ 0, we often use the concavity properties

(2.4) ωt(cs) ≤ cωt(s) , ωt(s1 + s2) ≤ 2ωt(s1) + 2ωt(s2) .


We finally recall the basic notions of solutions. A map u ∈ W 1,p(Ω,RN ) is aminimizer of the functional F [·] in (1.2) under the assumption (1.30)1 iff F [u] ≤F [u+w] whenever w ∈W 1,p

0 (Ω,RN ). Moreover, we say that v ∈W 1,p(Ω,RN ) is aweak solution to (1.6) under the assumptions (1.7)1 provided

∫

Ω

〈a(x, u,Du), Dϕ〉 dx = 0

holds whenever ϕ ∈W 1,p0 (Ω,RN ).

2.2. Other preliminaries. In the following we shall several times use the auxiliarymap

(2.5) V (z) = Vp(z) := (1 + |z|2) p−24 z (z ∈ RNn) .

A basic property of V (·), whose proof can be found for instance in [27], is in

Lemma 2.1. Let 1 < p <∞. There exists a constant c ≡ c(n,N, p) such that

c−1(1 + |z1|2 + |z2|2

) p−22 ≤ |V (z2)− V (z1)|2

|z2 − z1|2 ≤ c(1 + |z1|2 + |z2|2

) p−22

for any z1, z2 ∈ RNn.

We recall the definition of Morrey spaces.

Definition 2.1. With A ⊂ Rn denoting a bounded domain and q > 1, the Morreyspace Lq,µ(Ω,RNn) is defined as the space of those maps w ∈ Lq(Ω,RNn) such that

‖w‖qLq,µ(A) := sup%>0

%−µ

∫

Ω(x0,%)

|u(x)|q dx <∞ µ ∈ [0, n] ,

where A(x0, %) := A ∩B(x0, %), x0 ∈ A.

Our reference on Morrey spaces is [25, Chapter 2].Calderon-Zygmund coverings. For a cube Q0 ⊂ Rn denote with D(Q0) the

class of all dyadic sub-cubes of Q0, that is the class of those cubes, with sidesparallel to those of Q0, that have been obtained by a positive, finite number ofdyadic subdivisions of the cube Q0. Note that, in particular, Q0 6∈ D(Q0). Let usrecall a few simple properties of the class D(Q0). If Q1, Q2 ∈ D(Q0) then eitherthe two cubes are disjoint: Q1 ∩ Q2 = ∅, or one of the cubes contains the other:Q1 ⊆ Q2 or Q2 ⊆ Q1. We shall call Qp “a” predecessor of Q if Q has been obtainedfrom the the cube Qp through a finite number of subsequent dyadic subdivisions; weshall call Q ∈ D(Q0) “the” predecessor of Q if Q has been obtained by exactly onedyadic subdivision from the original cube Q. Denoting QR := (−R,R)n and Q+

R :=(−R,R)n−1 × (0, R), we shall need also to deal with Calderon-Zygmund coveringsinvolving rectangles of the form Q+

R and using the corresponding family of dyadicsub-rectangles - compare with (2.3). All the considerations are entirely similar,exchanging the words “dyadic sub-cube” by “dyadic sub-rectangle”, according tothe notation established in Section 2.1. These are obtained starting by an uppercube, and then proceeding in dyadic subdivisions using rectangles; see also [36].

The proof of the following result can be easily adapted from [10].

Proposition 2.1. Let Q0 ⊂ Rn be a cube and δ ∈ (0, 1). Assume that X ⊂ Y ⊂ Q0

are measurable sets satisfying the following conditions:(i) |X| < δ|Q0|(ii) if Q ∈ D(Q0) then |X ∩ Q| > δ|Q| =⇒ Q ⊂ Y , where Q denotes the

predecessor of Q.Then |X| < δ|Y |. The same result holds using rectangles instead of cubes.


Maximal Operators. Let Q0 ⊂ Rn be a cube. We shall consider, in thefollowing, the restricted maximal function operator relative to Q0, defined as

(2.6) M∗Q0

(f)(x) := supQ⊆Q0, x∈Q

−∫

Q

|f(x)| dx ,

whenever f ∈ L1(Q0), where Q denotes any cube contained in Q0, as long as x ∈ Q.We recall the following weak type (t, t) estimate for M∗

Q0, valid for any t ∈ [1,∞):

(2.7) |x ∈ Q0 : |M∗Q0

(f)(x)| ≥ λ| ≤ cW (n, t)λt

∫

Q0

|f(x)|t dx ∀ λ > 0 ,

which is valid for any f ∈ Lt(Q0); the constant cW depends only n and t; for thisand related issues we refer to [49, 29]. A standard consequence of the previousinequality is then

(2.8)∫

Q0

|M∗Q0

(f)(x)|t dx ≤ c(n, t)t− 1

∫

Q0

|f(x)|t dx , t > 1 .

Exactly as in (2.6) we can define a restricted maximal operator based on rectanglesas in (2.3) instead of cubes:

(2.9) M∗R0

(f)(x) := supR⊆R0, x∈R

−∫

Q

|f(x)| dx ,

where R0 is a given rectangle, and R is any other possible rectangle of the type(2.3), contained in R0. The properties of this maximal operator are the same asthose of M∗

Q0(only the constants involved in the maximal inequalities differ).

3. Fractional Sobolev spaces

Here we recall the definition of fractional Sobolev spaces together with a fewrelated properties that will be used later. Throughout we use the standard notationfrom [4], adapted to the situations we are going to deal with.

Definition 3.1. For a bounded open set A ⊂ Rn and k ∈ N, parameters θ ∈ (0, 1)and q ∈ [1,∞), we write w ∈ W θ,q(A,Rk) provided the following Gagliardo-typenorm is finite:

‖w‖W θ,q(A) :=(∫

A

|w(x)|q dx) 1

q

+(∫

A

∫

A

|w(x)− w(x)|q|x− x|n+θq

dx dx

) 1q

=: ‖w‖Lq(A) + [w]θ,q;A <∞ .(3.1)

In the following it will be convenient to put W 0,q ≡ Lq.For a vector valued function G : Ω → RNn, a vector e ∈ B1 ⊂ Rn and a real

number h ∈ R, we define the finite difference operator τe,h

τe,hG(x) ≡ τe,h(G)(x) := G(x+ he)−G(x) .

This makes sense whenever x, x + he ∈ A, an assumption that will be satisfiedwhenever we shall use τe,h. If e = es, s ∈ 1, ..., n, we write τs,h instead of τe,h.

Standing assumption. In the rest of the paper, we shall frequently use theoperators τs,h, with various restrictions on the size of h. Whenever we are dealingwith an integration domain which is an upper ball or an upper cube, we shall useτn,h, that is τs,h in the “direction orthogonal” to Γ1, always with h > 0.

Accordingly, here is a suitable boundary version of the well-known characteriza-tion of fractional Sobolev spaces via finite differences (Nikolski spaces). The proofis a simple variant of the standard one, presented for instance in [36, Lemma 2.5],taking into account the presence of a “lower boundary” ΓR.


Lemma 3.1. Let G ∈ Lq(Q+R,RNn), q ≥ 1, and assume that for θ ∈ (0, 1], M > 0,

0 < r < R, we have ∫

Q+r

|τs,hG|q dx ≤Mq|h|qθ

for every s ∈ 1, . . . , n and h ∈ R satisfying 0 < |h| ≤ d, where 0 < d ≤ min1, R−r is a fixed number. In the case s = n we only allow positive values of h. ThenG ∈ W b,q(Q+

% ,RNn) for every b ∈ (0, θ) and % < r. Moreover, there exists aconstant c ≡ c(n, q) independent of M and G, such that for every 0 < % < r thefollowing inequality holds true:

(3.2)∫

Q+%

∫

Q+%

|G(x)−G(x)|q|x− x|n+bq

dx dx ≤ c(Mqεq(θ−b)

θ − b +|Q+

R|εn+bq

∫

Q+R

|G|q dx),

where ε := minr − %, d.The Hausdorff dimension of the set of non-Lebesgue points of fractional Sobolev

maps can be estimated by the next result. This is a classical fact. A simple proofcan be obtained modifying the arguments given in [40, Section 4].

Proposition 3.1. Suppose that v ∈ W θ,q(Q+r ,Rk) for every 0 < r < d, where

d > 0 is a fixed number, and θ ∈ (0, 1], q ≥ 1, k ∈ N. Moreover, let

A :=

x ∈ Q+

d ∪ Γd : lim sup%0

−∫

B(x,%)∩Q+d

|v(y)− (v)B(x,%)∩Q+d|q dy > 0

,

B :=

x ∈ Q+

d ∪ Γd : lim sup%0

|(v)B(x,%)∩Q+d| =∞

.

Then dimH(A) ≤ n− θq and dimH(B) ≤ n− θq.The following lemma expresses a simple way to switch from a given decay esti-

mate for finite differences of V (G) to a decay estimate for the finite differences ofG, where V is as in (2.5).

Lemma 3.2. Let G ∈ Lp(Q+R,RNn), 2 ≤ p < ∞, s ∈ 1, . . . , n, and assume that

for θ ∈ (0, 1], M > 0 and 0 < r < R we have

(3.3)∫

Q+r

|τs,h(V (G))|2 dx ≤M2|h|2θ ,

for every 0 < |h| ≤ mind, R− r, where 0 < d ≤ min1, R− r is a fixed number.In the case s = n we only allow positive values of h. Then we have

∫

Q+r

|τs,hG|p dx ≤ c(n,N, p)M2|h|2θ .

For the proof of the previous lemma we refer to [17, Lemma 2.3]. We state asimple result whose easy proof is included for completeness.

Proposition 3.2. Let w ∈ W θ,q(Q+) where Q+ is an upper cube in the sense ofdefinition (2.2). Denoting x ≡ (x′, xn), we define i : Rn 3 (x′, xn) 7→ (x′,−xn) ∈Rn and

(3.4) w(x) :=

w(x) x ∈ B+

w(i(x)) x ∈ B− .

being the even extension of w to Q. Then ‖w‖W θ,q(Q) ≤ c(q)‖w‖W θ,q(Q). The sameholds replacing upper cubes with upper balls defined in (2.1).


Proof. Without loss of generality we assume that 0 < α < 1, other wise the state-ment is trivial. Moreover it is sufficient to prove that

(3.5) [w]θ,q;Q ≤ c[w]θ,q;Q+ .

To this aim, we split as follows:∫

Q

∫

Q


dx dx =∫

Q+

∫

Q++

∫

Q+

∫

Q−+

∫

Q−

∫

Q++

∫

Q−

∫

Q−.

By (3.4) the last term in the right hand side equals the first, and both of themare estimated by [w]qθ,q;Q+ ; moreover the second equals the third. Therefore itremains to estimate the second term. In turn we have, by the definition in (3.4)and changing variables, that∫

Q+

∫

Q−


dx dx =∫

Q+

∫

Q−

|w(x)− w(i(x))|q|x− x|n+θq

dx dx ≤ [w]θ,q;Q+ ,

where we used that |x − x| ≥ |x − i(x)|; the proof in the case of cubes is finished.The one in the case of balls is the same. ¤

We next recall the fractional Poincare’s inequality.

Proposition 3.3. If B ≡ B(x0, R) ⊂ Rn is a ball and w ∈W θ,q(B,Rk), then∫

B

|w − (w)B|q dx ≤ c(n)Rθq

∫

B

∫

B


dx dx

= c(n)Rθq[w]qθ,q;B .(3.6)

The same holds replacing the full ball B by an upper ball B+.

Proof. The proof of the first assertion can be retrieved from [40, Section 4]; theassertion concerning half balls easily follows from (3.6) via a reflection argumentwe briefly outline here. Without loss of generality we assume that the ball B iscentered at the origin. Consider the even reflection of w defined in (3.4), and notethat (w)B+ = (w)B. Therefore, using (3.5) for balls, we conclude with∫

B+|w − (w)B+ |q dx = 2

∫

B+|w − (w)B|q dx ≤ cRθq[w]qθ,q;B ≤ cRθq[w]qθ,q;B+ .

¤

4. Basic regularity results

4.1. Preliminary partial regularity. Here we recall some known and less knownboundary regularity results. Both in the case of solutions to Dirichlet problemsinvolving systems of the type (1.6), and in the variational case (1.1), solutions areof class C1,σ up to the boundary in the two dimensional case n = 2, for someσ > 0. This result has been proved in [5, Theorem 2] for elliptic systems under theassumptions (1.44) and (1.45). Note that in [5] the assumption that u is boundedis needed since the author treats non-homogeneous systems with a right-hand sidehaving critical growth in the gradient; such an assumption in not necessary in thecase of homogeneous systems considered in (1.6). As far as the variational case(1.1) is concerned, the everywhere regularity in the interior in the two-dimensionalcase has been proved in [36, Section 8] under assumptions (1.30)-(1.32) for someσ > 0; combining the techniques of [36] with the boundary technique of [5] leadsto the everywhere boundary regularity also for (1.1); compare with [6]. Therefore,when dealing with the singular sets at the boundary, we can focus on n ≥ 3 (andstill N ≥ 2).

We now switch to the partial Holder continuity of solutions, to state two resultsthat will be crucial for treating the low dimensional case n ≤ p + 2. Again we


start from the following result, which is actually an immediate consequence of [5,Theorem 1]:

Theorem 4.1. Let u ∈ W 1,p(Ω,RN ) with p ∈ [n − 2, n), be a a solution to (1.6),under the assumptions (1.3), and (1.44)-(1.45). There exists an open subset Ωu ⊂Ω, a relatively open subset Ou ⊂ ∂Ω, and a number

(4.1) µ1 ≡ µ1(n,N, p, L/ν) ∈ (2, n) ,

such that

(4.2)

dimH(Ω \ (Ωu ∪ Ou)) ≤ n− p Du ∈ Lp,µ1(Ωu ∪ Ou,RNn)

u ∈ C0,λ1(Ωu ∪ Ou,RN ) λ1 := 1− (n− µ1)/p .

The last theorem tells us in particular that, since p > 1, Hn−1-almost everypoint x0 ∈ ∂Ω is such that u is Holder continuous in a neighborhood of x0. Asimilar fact holds in the variational case:

Theorem 4.2. Let u ∈ W 1,p(Ω,RN ) with p ∈ [n − 2, n), be a solution to (1.1),under the assumptions (1.3) and (1.30)-(1.32). There exists an open subset Ωu ⊂ Ω,a relatively open subset Ou ⊂ ∂Ω, and a number µ1 ∈ (2, n), exhibiting the samedependence as in (4.1), such that (4.2) holds.

The interior version of this result, i.e. with Ωu ∪ Ou replaced by Ωu ⊂ Ω, hasbeen obtained in [36, Theorem 8.1]. The up-to-the-boundary version stated inTheorem 4.2 can be obtained by combination of the interior variational method of[36, Theorem 8.1] and the technique of [5, Theorem 1], via the flattening-of-the-boundary method [11, 5, 33] used for treating boundary value problems as in (1.6);compare with [39]. This result and its proof can be found in [6].

4.2. Higher integrability exponents. Here we collect a few basic regularity re-sults that hold for basic elliptic systems. Let v ∈W 1,p(Ω,RN ) be a solution to theelliptic system

(4.3) divA(Dv) = 0 in Ω ,

where the C2-vector field A(·) satisfies the conditions

(4.4) ν(1 + |z|2) p−22 |λ|2 ≤ 〈Az(z)λ, λ〉 , |Az(z)| ≤ L(1 + |z|2) p−2

2

whenever z, λ ∈ RNn; here Ω denotes a bounded open domain in Rn, and notnecessarily the one from (1.1)-(1.6). Then it follows that there exists a higherintegrability exponent 2δi > 0 depending only on n,N, p, L/ν

(4.5) v ∈W 1,q1loc (Ω,RNn) , with q1 :=

np

n−2 + 2δi if n > 2any finite number if n = 2 .

See [33] for an up-dated presentation. We now set

(4.6) q := q1 − δi , and hence q =np

n− 2+ δi for n > 2 .

Moreover, it follows that, whenever Q is a a cube or half-cube - i.e. a rectangle asin (2.3) - such that 2Q ⊂ Ω and |Q| ≤ 10n|B1|, we have

(4.7)(−∫

Q

(|Dv|+ χ|v|)q1 dx

) 1q1 ≤ c

(−∫

λQ

(1 + |Dv|+ χ|v|)p dx

) 1p

,

whenever λ ∈ (1, 2), where c ≡ c(n,N, p, L/ν, λ), and either χ = 0 or χ = 1. Wecan now disclose the identity of the exponent ns appearing in (1.28) and (1.31); forn ≥ 3, which is the case of interest here, we set

(4.8) ns := n− (n− 2)2δi2p+ (n− 2)δi

< n .


Of course we may always assume δi small enough to make ns > 0. The dependencestated in (1.28) follows at once recalling that δi depends on n,N, p, L/ν. Note thatthe following identity holds:

(4.9) ns = 2(q

p

)′=

2qq − p .

shall deal with a non-negative g(·) ∈ Lns and Du ∈ Lq, it will be possible toestimate, via Young’s inequality We finally recall a boundary version of Gehring’slemma from [2, 23]. Let v ∈ u+W 1,p

0 (Q,RN ) be a minimizer of the functional F [·]in (1.2), under the only assumption (1.30)1, with u ∈ W 1,p(Q,RN ) and Ω ≡ Q.Here Q is a cube or a rectangle. Then there exists q2 ≡ q2(n,N, p, L/ν) ∈ (p, q),and constant c ≡ c(n,N, p, L/ν), such that

(4.10)(−∫

Q

|Dv|q2 dx

) 1q2 ≤ c

(−∫

Q

(1 + |Dv|p) dx) 1

p

+ c

(−∫

Q

(1 + |Du|q2) dx) 1

q2

.

minimizer of the functional F [·] in (1.2), again under the sole assumption (1.30)1,and such that v ≡ 0 on Q ∩ xn = 0, in the sense of traces.

4.3. Basic boundary regularity. We shall consider the following problem, wherethe Dirichlet boundary condition is prescribed only on a part of the boundary:

(4.11)

divB(x,Dv) = 0 in A

v = 0 on ∂A ∩ Γ1 .

Here A is a bounded, Lipschitz-domain such that B+R ⊂ A ⊆ B+

2R, where BR is aball centered on xn = 0 and R ∈ (0, 1]. The vector field B : A × RnN → RnN ,satisfies the following assumptions:

(4.12)

ν(1 + |z|2) p−2

2 |λ|2 ≤ 〈Bz(x, z)λ, λ〉 , |Bz(x, z)| ≤ L(1 + |z|2) p−22

|Bx(x, z)| ≤ Lγ(x)(1 + |z|2) p−12 ,

for any z, λ ∈ RnN , and x ∈ A, where p ≥ 2, 0 < ν ≤ L and γ(·) is a non-negativemeasurable function.

We are interested in the existence and integrability of second order derivativesup to ∂A∩Γ1. This involves a standard difference quotients argument for boundaryproblems [11]. We shall give later a formal sketch of the proof in order to emphasizethe a priori estimates involved, whose exact form will play a crucial role in thefollowing.

Theorem 4.3. Let v ∈ W 1,p(A,RN ) be a weak solution to (4.11), under the as-sumptions (4.12). In addition assume that

(4.13) γ(·) ∈ Lns(A) and |Du| ∈ Lq(A) .

Then V (Dv) ∈ W 1,2(Q+2d,RNn). Moreover, whenever V ∈ RNn, for a constant

c ≡ c(n,N, p, L/ν) we have that∫

B+R/2

|D(V (Dv))|2 dx ≤ c

R2

∫

B+R

|V (Dv)− V |2 dx+ c

∫

B+R

γ2(1 + |Dv|)p dx .

The map V (·) has been defined in (2.5) and ns in (4.8). The last result is ofcourse the boundary version of a similar interior Caccioppoli’s inequality.

Theorem 4.4. Let v ∈W 1,p(BR,RN ) be a weak solution to the system

div B(x,Dv) = 0 in BR ,


under the assumptions (4.12). In addition, assume that (4.13) holds with A replacedby BR. Then V (Dv) ∈ W 1,2(BR/2,RNn). Moreover, whenever V ∈ RNn, for aconstant c ≡ c(n,N, p, L/ν) we have that

(4.14)∫

BR/2

|D(V (Dv))|2 dx ≤ c

R2

∫

BR

|V (Dv)−V |2 dx+c∫

BR

γ2(1+|Dv|)p dx .

Sketch of the proofs of Theorems 4.3-4.4. Here we give a formal sketch; allthe arguments can be made rigorous by using the difference quotient methods asoutlined before Theorem 4.3. We begin with the interior estimate, starting by theweak formulation∫

BR

〈B(x,Dv), Dϕ〉 dx = 0 ∀ϕ ∈ C∞0 (BR,RN ).

For s = 1, . . . , n we substitute ϕ←→ Dsϕ and integrate by parts, thereby obtaining

(4.15)∫

BR

〈Bz(x,Dv)DDsv,Dϕ〉 dx = −∫

BR

〈Bx(x,Dv), Dϕ〉 dx .

Now take ϕ = η2(Dsv − vs), where η ∈ C∞0 (BR) such that η ≡ 1 on BR/2 and|Dη| ≤ cR−1, moreover v ∈ RNn; of course vs = (vα

s )1≤α≤N - this is the only pointwhere one really needs difference quotients to be rigorous. Therefore, using growthand ellipticity assumptions in (4.12) we obtain in a standard way

(I) :=∫

BR

(1 + |Dv|2) p−22 |DDsv|2η2 dx

≤ c

∫

BR

(1 + |Dv|2) p−22 |DDsv||Dsv − vs||Dη| dx

+c∫

BR

γ(1 + |Dv|2) p−12 [2η|Dη||Dsv − vs|+ η2|DDsv|] dx

=: (II) + (III) + (IV ) .

By using Young’s inequality with ε ∈ (0, 1) one estimates

(II) ≤ ε(I) +c(ε)R2

∫

BR

(1 + |Dv|2) p−22 |Dsv − vs|2 dx ,

(III) ≤ c

R2

∫

BR

(1 + |Dv|2) p−22 |Dv − vs|2 dx+ c

∫

BR

γ2(1 + |Dv|2) p2 dx ,

(IV ) ≤ ε(I) + c(ε)∫

BR

γ2(1 + |Dv|)p dx ,

and all the constants depend also n,N, p, L/ν. Choosing ε small enough, mergingthe last four inequalities, and finally summing up over s ∈ 1, . . . , n yields

∫

BR

(1 + |Dv|2) p−22 |D2v|2η2 dx ≤ c

R2

∫

BR

(1 + |Dv|2) p−22 |Dv − v|2 dx

+c∫

BR

γ2(1 + |Dv|)p dx .

Now (4.14) follows by observing that

|D(V (Dv))|2 . (1+ |Dv|2) p−22 |D2v|2 , (1+ |Dv|2) p−2

2 |Dv− v|2 . |V (Dv)−V (v)|2,and taking v such that V = V (v). This last fact is possible since, in view of Lemma2.1, the map V (·) is a local bi-Lipschitz bijection of RNn.


Passing to the boundary case, that is to Theorem 4.3, the previous argumentstill works in the tangential directions s ∈ 1, . . . , n− 1 as in this case Dsv ≡ 0 onΓ1 ∩ ∂A, therefore we have∫

B+R/2

(1 + |Dv|2) p−22 |DD′v|2 dx ≤ c

R2

∫

B+R

|V (Dv)− V |2 dx

+c∫

B+R

γ2(1 + |Dv|)p dx ,(4.16)

where D′ ≡ (D1, . . . , Dn−1) denotes the tangential gradient - with respect to Γ1.On the other hand, from the interior result case, it follows that the solution v isdifferentiable in the interior. In particular we have that V (Dv) ∈ W 1,2

loc (A,RNn)and therefore also Dv ∈ W 1,2

loc (A,RNn). Therefore v is a so called strong solution,and we can differentiate the system getting

N∑

β=1

n∑

i,j=1

(Bαi )zβ

j(x,Dv)Dijv

β = −n∑

i=1

(Bαi )x(x,Dv) , α = 1, ..., N ,

almost everywhere in A. The idea is to get double normal derivatives via thosecontaining a normal component using the equation, i.e.

N∑

β=1

(Bαn )zβ

n(x,Dv)Dnnv

β = −N∑

β=1

n∑

i,j=1

(Bαi )zβ

j(x,Dv)Dijv

β −n∑

i=1

(Bαi )x(x,Dv) ,

for every α = 1, ..., N , where∑

denotes the sum with respect to all terms withindexes i, j not simultaneously equal to n. Then we multiply the previous relationby Dnnv

α and sum over α. Using growth conditions and ellipticity we get

ν(1+ |Dv|2) p−22 |Dnnv|2 ≤ L(1+ |Dv|2) p−2

2 |DD′v||Dnnv|+ γ(1+ |Dv|2) p−12 |Dnnv| ,

and by Young’s inequality

(1 + |Dv|2) p−22 |Dnnv|2 ≤ c(1 + |Dv|2) p−2

2 |DD′v|2 + cγ2(1 + |Dv|2) p2 .

Using this pointwise inequality in combination with (4.16) gives the proof of The-orem 4.3.

As mentioned above, the previous argument is in certain parts formal, must canbe easily made rigorous via the use of a difference quotients argument; one takes|h|−2τs,−h[η2τs,h(v − xsvs)] with s ∈ 1, . . . , n and v ∈ RNn, as a test function,and then proceeds as above using discrete integration-by-parts and letting h → 0.This leads to the terms (I), . . . , (IV ) arising after (4.15) that at this point can beestimated as above; the existence of interior full second derivatives follows togetherwith (4.14). In a second step one takes one takes |h|−2τs,−h[η2τs,h(v − xsvs)] withs 6= n and gets the existence of second order derivatives with a tangential compo-nent together with (4.16). Then, again proceeding exactly as above after (4.16),one estimates double normal derivatives via second derivatives with a tangentialcomponent and Theorem 4.3 follows. We refer to [7, Sections 4.1-4.2] for relatedcomputations.

We remark that throughout the proof the assumption (4.13) has been used in or-der to ensure the integrability of γ2|Du|p, since, by Young’s inequality and (4.9), wecan estimate γ2|Du|p ≤ cγns + |Du|q, and, also at the level of difference quotients,all the terms involved are finite.

We finally conclude with the boundary version of inequality (4.7).

Theorem 4.5. Let v ∈ W 1,p(A,RN ) be a weak solution to (4.11), where A ≡ Q+R

for some R > 0, and the vector field B(·) satisfies assumptions (4.12)1 and isindependent of x, i.e. B(x,Du) ≡ B(Du). Then, whenever Q+ is an upper cube


- therefore centered on xn = 0 - and such that (2Q)+ ⊂ Q+R and |Q| ≤ 10n, it

holds that

(4.17)(−∫

Q+(|Dv|+ χ|v|)q1 dx

) 1q1 ≤ c

(−∫

(λQ)+(1 + |Dv|+ χ|v|)p dx

) 1p

,

whenever λ ∈ (1, 2] and χ ∈ 0, 1, where c ≡ c(n,N, p, L/ν, λ), and q1 has beendefined in (4.5).

The proof of Theorem 4.5 can be easily obtained using Theorem 4.3 with γ(·) ≡ 0,and the using Sobolev embedding theorem.

5. Comparison lemmas

In the following we shall state several crucial comparison lemmas - first in theinterior case, and then up to the boundary. In the whole section we shall alwaysconsider a radius R ≤ 1.

5.1. Interior balls estimate. We consider a function F (·) satisfying (1.30)-(1.32)in Ω×RN×RNn, where Ω ⊂ Rn is as usual an open bounded domain. We take a ballB with radius R, and such that 4B ⊂ Ω. Next, we fix a smooth, positive, radiallysymmetric convolution kernel φ supported in B1, and satisfying

∫B1φdx = 1, and

define

F (x, z) :=∫

B1

F (x+Rx, (u)B, z)φ(x) dx

=1Rn

∫

BR(x)

F (x, (u)B, z)φ(x− xR

)dx ,(5.1)

and we recall that(u)B := −

∫

B

u(x) dx .

With the above definitions, and using (1.30), it is easy to see that the new energydensity F (·) satisfies the following conditions:

(5.2)

ν|z|p ≤ F (x, z) ≤ L(1 + |z|2) p2

ν(1 + |z|2) p−22 |λ|2 ≤ 〈Fzz(x, z)λ, λ〉 ≤ L(1 + |z|2) p−2

2 |λ|2|F (x1, z)− F (x2, z)| ≤ L|x1 − x2|σ(1 + |z|2) p

2 ,

where the notation is the same as in (1.30), but here x, x1, x2 ∈ B. Moreover,applying a standard result on convex functions - see for instance [24] - we recallthat (5.2) implies that

(5.3) |Fz(x1, z)− Fz(x2, z)| ≤ c(n,L/ν)|x1 − x2|σ2 (1 + |z|2) p−12 .

In addition we remark - compare with [13, Proposition 2.32] - that (5.2)1,2 imply

(5.4) |Fz(x, z)| ≤ c(L)(1 + |z|2) p−12 .

Lemma 5.1. Let u ∈ W 1,qloc (Ω,RN ) be a local minimizer of the functional F [·] in

(1.2), under the assumptions (1.30)-(1.32), and where q has been defined in (4.6).Let v ∈ u+W 1,p

0 (B,RN ) be the unique solution to

minw

∫

B

F (x,Dw) dx w ∈ u+W 1,p0 (B,RN ) ,

where F (·) is defined in (5.1), and assume that

(5.5)τγ

τ − β ≤ q where 2 ≤ τ ≤ p .


Then there exists a positive constant c depending only on n,N, p, L/ν, σ, such that

(5.6)∫

B

|V (Du)− V (Dv)|2 dx ≤ c∫

2B

(1 + |Du|q + gns) dxRσv

holds, where

(5.7) σv ≡ σv(τ) := min

2α, q + p− 2γ,τβ

τ − β.

Proof. The minimality of v allows to use the Euler-Lagrange system div Fz(x,Dv) =0; therefore

(5.8)∫

B

〈Fz(x,Dv), Du−Dv〉 dx = 0 .

Note that (5.8) holds by (5.4). We now have

(5.9)∫

B

|Dv|q dx ≤ c∫

B

(1 + |Du|q) dx ,

where the constant c only depends on n,N, p, L/ν and σ; in particular it followsthat v ∈ W 1,q(B,RN ). This fact follows applying Proposition 8.6 below to thesystem div Fz(x,Dv) = 0 and taking into account (5.2) and (5.3). Now, usingTaylor’s formula together with (5.2)2 it is possible to find a positive constant c ≡c(n,N, p, L/ν) such that, for all z1, z2 ∈ RNn and x ∈ Ω, it holds that

(5.10)1c(1 + |z1|2 + |z2|2)

p−22 |z2 − z1|2 ≤ F (x, z2)− F (x, z1)− 〈Fz(x, z1), z2 − z1〉

and using this last fact in combination with Lemma 2.1 we conclude with1c

∫

B

|V (Du)− V (Dv)|2 dx

≤∫

B

(F (x,Du)− F (x,Dv)− 〈Fz(x,Dv), Du−Dv〉

)dx

for a positive constant c ≡ c(n,N, p, L/ν). Therefore (5.8) yields

(5.11)1c

∫

B

|V (Du)− V (Dv)|2 dx ≤∫

B

(F (x,Du)− F (x,Dv)

)dx

and also1c

∫

B

|V (Du)− V (Dv)|2 dx ≤∫

B

(F (x,Du)− F (x,Dv)

)dx

+∫

B

(F (x, v,Dv)− F (x, u,Du)

)dx .

Indeed observe that the last integral is non-negative due to the minimality of u.Keeping in mind the definition in (5.1), the last inequality can be re-written as

1c

∫

B

|V (Du)− V (Dv)|2 dx ≤∫

B1

∫

B

[(F (x+Rx, (u)B, Du)− F (x, u,Du)

)

−(F (x+Rx, (u)B, Dv)− F (x, u,Dv)

)]dxφ(x) dx

+∫

B

(F (x, v,Dv)− F (x, u,Dv)

)dx ,(5.12)

and we recall that x is the integration variable in B1. Letting

G(x, z) ≡ GR,x(x, z) := F (x+Rx, (u)B, z)− F (x, u(x), z)

we again re-write (5.12) as1c

∫

B

|V (Du)− V (Dv)|2 dx ≤∫

B1

∫

B

(G(x,Du)−G(x,Dv)

)dxφ(x) dx


+∫

B

(F (x, v,Dv)− F (x, u,Dv)

)dx := (I) + (II) .(5.13)

We now proceed estimating the last two terms. As for the first, we write

G(x,Du)−G(x,Dv) = 〈∫ 1

0

Gz(x, tDv + (1− t)Du) dt,Du−Dv〉

= 〈∫ 1

0

Fz(x+Rx, (u)B, tDv + (1− t)Du)−Fz(x, u(x), tDv + (1− t)Du) dt,Du−Dv〉 ,

and therefore we also have

G(x,Du)−G(x,Dv) = 〈∫ 1

0

Fz(x+Rx, (u)B, tDv + (1− t)Du)−Fz(x, (u)B, tDv + (1− t)Du) dt,Du−Dv〉 dt

+〈∫ 1

0

Fz(x, (u)B, tDv + (1− t)Du)−Fz(x, u(x), tDv + (1− t)Du) dt,Du−Dv〉 .

Using (1.30)4,5 together with the previous equality yields, and recalling that ωβ(·) ≤ωα(·) as we are assuming α ≤ β, we have

(I)

≤ cRα

∫

B1

∫

B

(g(x) + g(x+Rx))(1 + |Du|+ |Dv|)p−1|Du−Dv| dxφ(x)dx

+c∫

B

ωα(|u− (u)B|)(1 + |Du|+ |Dv|)γ−1|Du−Dv| dx := (III) + (IV ) .(5.14)

Let us estimate (III). Using (4.9) and Young’s inequality with ε > 0, we have

(III) = cRα

∫

B1

∫

B

(g(x) + g(x+Rx))(1 + |Du|+ |Dv|) p2

×(1 + |Du|+ |Dv|) p2−1|Du−Dv| dxφ(x)dx

≤ ε

∫

B

(1 + |Du|2 + |Dv|2) p−22 |Du−Dv|2 dx

+c(ε)R2α

∫

B1

∫

B

(g(x) + g(x+Rx))2(1 + |Du|+ |Dv|)p dxφ(x)dx

≤ ε

∫

B

(1 + |Du|2 + |Dv|2) p−22 |Du−Dv|2 dx

+c(ε)R2α

∫

B1

∫

B

(g(x) + g(x+Rx))ns dxφ(x)dx

+c(ε)R2α

∫

B

(1 + |Du|+ |Dv|)q dx

≤ cε

∫

B


+c(ε)R2α

∫

2B

gns dx+ c(ε)R2α

∫

B

(1 + |Du|+ |Dv|)q dx .(5.15)

We have used∫

B1

∫

B

(g(x) + g(x+Rx))ns dxφ(x)dx ≤ 2ns

∫

B1

∫

2B

g(x)ns dxφ(x)dx

= 2ns

∫

2B

gns dx ,(5.16)


and finally, we have used Lemma 2.1 to estimate

(5.17)∫

B

(1 + |Du|2 + |Dv|2) p−22 |Du−Dv|2 dx ≤ c

∫

B

|V (Du)− V (Dv)|2 dx .

We proceed with the estimation of (IV ). In the following without loss of generalitywe shall assume that γ > p/2. Indeed, notice that if (1.30) holds for a certain valueγ ≤ 0, then it still holds for any other larger value of γ. Now, Young’s inequalityand (5.17) yield

(IV ) ≤ cRα

∫

B

ωα(|u− (u)B|)(1 + |Du|+ |Dv|)γ− p2

×(1 + |Du|+ |Dv|) p2−1|Du−Dv| dx

≤ ε

∫

B


+c(ε)∫

B

[ωα(|u− (u)B|)]2(1 + |Du|+ |Dv|)2γ−p dx .

In turn we have, again by Young’s inequality∫

B

[ωα(|u− (u)B|)]2(1 + |Du|+ |Dv|)2γ−p dx

≤ cRσv

∫

B

R−(2γ−p+σv)[ωα(|u− (u)B|)]2(2γ−p+σv)

σv dx

+cRσv

∫

B

(1 + |Du|2γ−p + |Dv|2γ−p)2γ−p+σv

2γ−p dx

≤ cRσv

∫

B

R−(2γ−p+σv)|u− (u)B|2γ−p+σv dx

+cRσv

∫

B

(1 + |Du|+ |Dv|)2γ−p+σv dx

≤ cRσv

∫

B

(1 + |Du|+ |Dv|)2γ−p+σv dx

≤ cRσv

∫

B

(1 + |Du|+ |Dv|)q dx .(5.18)

Of course we have used that 2γ − p + σv ≤ q. Note that in the last chain ofinequalities we used Poincare’s inequality, and since σv ≤ 2α we estimated asfollows:

[ωα(|u− (u)B|)]2(2γ−p+σv)

σv ≤ [ωα(|u− (u)B|)]2(2γ−p+σv)

2α ≤ |u− (u)B|2γ−p+σv .

Therefore we conclude with

(IV ) ≤ ε∫

B

|V (Du)− V (Dv)|2 dx+ c(ε)Rσv

∫

B

(1 + |Du|+ |Dv|)q dx .

Connecting this last estimate with the one found for (III) in (5.15), and finallyusing that R ≤ 1 and σv ≤ 2α, we have, for every ε ∈ (0, 1), that

(I) ≤ (III) + (IV ) ≤ 2ε∫

B


+c(ε)Rσv

∫

B

(1 + |Du|+ |Dv|)q dx+ c(ε)Rσv

∫

2B

gns dx .(5.19)

We now come to the estimate of (II), which has been defined in (5.13). Here theparameter τ ∈ [2, p] introduced in (5.5) comes into the play. First let us observethat by Lemma 2.1 it easily follows that

|Du−Dv|p+|Du−Dv|2 ≤ 2(1+|Dv|2+|Du|2) p−22 |Du−Dv|2 ≤ c|V (Du)−V (Dv)|2,


and therefore, keeping in mind the elementary inequality sτ ≤ s2 + sp for s ≥ 0,there exists a constant c ≡ c(n,N, p, L/ν) such that

(5.20) |Du−Du|τ ≤ c|V (Du)− V (Dv)|2 for every τ ∈ [2, p] .

Using this fact, (1.30)3 and Young’s and Poincare’s inequalities, we have that

(II) ≤ c

∫

B

ωβ(|u− v|)(1 + |Dv|)γ dx

≤ c

∫

B

R−β |u− v|β(1 + |Dv|)γ dxRβ

≤ ε

∫

B

R−τ |u− v|τ dx+ c(ε)Rτβ

τ−β

∫

B

(1 + |Dv|) τγτ−β dx

≤ cε

∫

B

|Du−Dv|τ dx+ c(ε)Rτβ

τ−β

∫

B

(1 + |Dv|) τγτ−β dx

≤ cε

∫

B

|V (Du)− V (Dv)|2 dx+ c(ε)Rσv(τ)

∫

B

(1 + |Dv|)q dx .(5.21)

Note that in the last inequality we have used (5.5) and the definition of σv ≡ σv(τ)in (5.7) to estimate Rτβ/(τ−β) ≤ Rσv(τ) since R ≤ 1. Merging (5.21) with thepreceding estimate for (I) in (5.19), and eventually plugging the resulting estimatein (5.13) we get∫

B

|V (Du)− V (Dv)|2 ≤ cε∫

B


+c(ε)Rσv

∫

B


∫

2B

gns dx ,(5.22)

where c ≡ c(n,N, p, L/ν). At this stage, choosing ε = (2c)−1 in the last estimate,and finally using (5.9) in the resulting inequality yields (5.6) and therefore theassertion of the Lemma. ¤

Finally, towards the proof of Theorem 1.8, we give a version of the previouslemma suited for functionals of the type

(5.23) F4[w] :=∫

Ω

f(x,Dw) + h(x,w) dx ,

where, as in Theorem 1.8, the integrand f(·) is a Caratheodory function satisfying(1.41), while h(·) is again a Caratheodory function satisfying (1.42). We define theregularized integrand f(·) as in (5.1), but with F (·) replaced by f(·) and, whendealing with structures as in Theorem 1.8 the function F (·) will be consequentlyreplaced by

(5.24) f(x, z) :=∫

B1

f(x+Rx, z)φ(x) dx .

Note that in (5.24), distinct from (5.1), the average (u)B is not involved.

Lemma 5.2. Let u ∈ W 1,qloc (Ω,RN ) be a local minimizer of the functional F4[·],

where f(·) satisfies (1.41), while h(·) satisfies (1.42), and q has been defined in(4.6). For 4B ⊂ Ω, let v ∈ u+W 1,p

0 (B,RN ) be the unique solution to the problem

minw

∫

B

f(x,Dw) dx w ∈ u+W 1,p0 (B,RN ) ,

and assume that (5.5) holds. Then there exists a positive constant c depending onlyon n,N, p, L/ν, σ, such that

(5.25)∫

B

|V (Du)− V (Dv)|2 dx ≤ c∫

2B

(1 + |Du|q + |u|q + gns) dxRσ0


holds, where

(5.26) σ0 := min

2α,tβ

t− β.

Proof. Let us first notice that (5.9) follows in this case applying applying Proposi-tion 8.6 below to the system div fz(x,Dv) = 0 and taking into account that (5.2)and (5.3) are satisfied by f(·) as well. We proceed as for Lemma 5.1, and using theminimality of u and the structure of the functional F4[·], we arrive at

1c

∫

B

|V (Du)− V (Dv)|2 dx ≤∫

B

(f(x,Du)− f(x,Dv)

)dx

+∫

B

(f(x,Dv)− f(x,Du)

)dx

+∫

B

(h(x, v)− h(x, u)

)dx .

Therefore, letting this time

G(x, z) ≡ GR,x(x, z) := f(x+Rx, z)− f(x, z) ,

we get the following analog of (5.13):

1c

∫

B

|V (Du)− V (Dv)|2 dx ≤∫

B1

∫

B

(G(x,Du)−G(x,Dv)

)dxφ(x) dx

+∫

B

(h(x, v)− h(x, u)

)dx := (I) + (II) .(5.27)

The estimate of (I) can be done exactly as in Lemma 5.1 and is

(I) ≤ 2ε∫

B


+c(ε)Rσ0

∫

B

(1 + |Du|+ |Dv|)q dx+ c(ε)Rσ0

∫

2B

gns dx

≤ 2ε∫

B


+c(ε)Rσ0

∫

B

(1 + |Du|)q dx+ c(ε)Rσ0

∫

2B

gns dx ,

for ε ∈ (0, 1); here we applied (5.9). The estimate of (II) is via Young’s inequality

(II) ≤ c

∫

B

|u− v|β(1 + |u− v|+ |u|)γ dx

≤ cε

∫

B

R−t|u− v|τ + c(ε)Rτβ

τ−β

∫

B

(1 + |u− v| τγτ−β + |u| τγ

τ−β ) dx

≤ cε

∫

B


τ−β

∫

B

(1 + |u− v|q + |u|q) dx

≤ cε

∫

B


τ−β

∫

B

(1 + |Du|q + |u|q + |Dv|q) dx

≤ cε

∫

B

|V (Du)− V (Dv)|2 dx+ c(ε)Rτβ

τ−β

∫

B

(1 + |Du|q + |u|q) dx .

We have just used Poincare’s inequality and that (5.5) and (5.9) hold, and in thelast line we used (5.20). Merging the estimates for (I) and (II) with (5.27), andchoosing ε small enough finally yields (5.25). ¤


5.2. Boundary upper balls estimate. Now we want to give a suitable boundaryversion of Lemma 5.1. The proof of the modified lemma in itself will be quitesimilar to the original one, and indeed we shall restrict ourselves to give the suitablemodifications. On the other hand, before giving such a proof, at some stages weface a certain number of delicate extra technicalities due to the fact that we areworking with the boundary situation. In particular we shall need a version of certainglobal integrability estimates in which the stability with respect to the boundarycurvatures will play a crucial role - see Proposition 8.6 below.

We shall assume to deal with a solution u ∈W 1,p(Q+1 ,RN ) of the minimization

problem(5.28)

∫

Q+1

F (x, u,Du) dx ≤∫

Q+1

F (x,w,Dw) dx ∀ w ∈ u+W 1,p0 (Q+

1 ,RN )

w ≡ u ≡ 0 on Γ1 .

The hypotheses on F (·) are those of Section 5.1, that is (1.30)-(1.32). At this stagewe are assuming to work with a Caratheodory function F : Q+

1 × RN × RNn → Rsatisfying (1.30) with Ω ≡ Q+

1 and for a suitable non-negative function g(·) ∈Lns(Q+

1 ), again defined in Q+1 . In particular this means that

(5.29) |Fz(x1, y, z)− Fz(x2, y, z)| ≤ (g(x1) + g(x2))|x1 − x2|α(1 + |z|2) p2

for almost every (x1, x2) ∈ Q+1 ×Q+

1 , and every choice (y, z) ∈ RN × RNn.In the following, we consider a ball BR centered on the hyperplane xn = 0,

together with his upper ball B+R ; the ball will be such that B4R ⊂ Q1, so that, in

particular, (B4R)+ ⊂ (Q1)+.Now, we first extend F (·) to a new function defined on Q1 × RN × RNn, still

denoted by F ; this is done as usual by an even reflection. Denoting x ≡ (x′, xn),and i : Rn 3 (x′, xn) 7→ (x′,−xn) ∈ Rn, we let

(5.30) F (x, y, z) :=

F (x, y, z) x ∈ Q+1

F (i(x), u, z) x ∈ Q−1 .

Note that, since Γ1 is a null set, and we are considering (5.29) which is naturallydefined almost everywhere since it involves the function g(·), the values on Γ1 ofx 7→ F (x, ·, ·) are not relevant. Accordingly, we extend the function g(·) appearingin (5.29) - again denoting the resulting extension with g(·) - by

(5.31) g(x) :=

g(x) x ∈ Q+1

g(i(x)) x ∈ Q−1 .

Obviously, we still have that g ∈ Lns(Q1). It is easy to check, and we leave it to thereader, that the extended function F (·) still satisfies the conditions (1.30)-(1.32) onQ1 × RN × RNn; in particular it satisfies (5.29) for every choice x1, x2 ∈ Q1.Small regular domains. In the proof of Lemma 5.1 we used inequality (5.9) inorder to control the integrability of the comparison minimizer v via the integrabilityof the original minimizer u. We now would like to do this on upper balls, but this isimpossible, since upper balls are only Lipschitz domains, while estimate (5.9) worksin general if the underlying domain is regular enough, for instance of class C1,σ forsome σ > 0. We shall therefore pass from B+

R to B+2R via an intermediate C1,α

domain. We shall without loss of generality assume that the ball BR is centeredat the origin, the general cases following by translation. We determine a startingdomain A1 - for instance via an axial symmetry argument - which is C1,α regularand satisfies

(5.32) B+1 ⊂ A1 ⊂ B+

2


such that ∂A1 ∩ Γ2 = Γ1. Then we define

(5.33) AR := Rx ∈ Rn : x ∈ A1 R > 0 .

Scaling (5.32) yields

(5.34) B+R ⊂ AR ⊂ B+

2R

and we also have that ∂AR ∩ Γ2R = ΓR.

Remark 5.1 (Flattening of the boundary). For later convenience we recall herethe flattening of the boundary procedure for C1,α domains where 0 < α ≤ 1; sucharguments are standard bur we have to put them here in the form needed later, agood reference at this stage is [32]. The domain A1 is C1,α regular in the sense thatthere exists a function Φ ∈ C1,α(Rn) such that A1 = x ∈ Rn : Φ(x) < 0 andRn \ A1 = x ∈ Rn : Φ(x) > 0, with DΦ(x0) 6= 0 whenever x0 ∈ ∂A1. Let us fixx0 ∈ ∂Ω; up to a rigid motion there is no loss of generality in assuming that x0 = 0and ∂Ω meets xn = 0 tangentially. Then, using the implicit function theoremwe find R1 > 0 depending on the point x0, and a C1,α-function φ : Rn−1 → R -φ ≡ φ(x′) - such that φ(0) = 0, Dφ(0) = 0 and

A1(0, R1) := A1 ∩Q(0, R1) = x = (x′, xn) ∈ Q(0, R1) : xn > φ(x′)and

Γ(0, R1) := ∂A1 ∩Q(0, R1) = x = (x′, xn) ∈ Q(0, R1) : xn = φ(x′) .We define the C1,α-diffeomorphisms ϕ,ψ : Rn → Rn as

(5.35)

ϕj(x) = xj

ϕn(x) = xn − φ(x′)and

ψj(x) = xj

ψn(x) = xn + φ(x′)

where j ∈ 1, . . . , n − 1. These are C2 maps, and of course ψ = ϕ−1. Moreover‖Dϕ− I‖L∞ + ‖Dψ − I‖L∞ ≤ ε.(5.36) ‖D2ϕ‖L∞ + ‖D2ψ‖L∞ ≤M1

holds for some M1 ≥ 1, and detϕ = detψ = 1. In addition, we have

ϕ(A1 ∩Q(0, R1)) ⊂ Q+1 ϕ(∂A1) ⊂ Q1 ∩ xn = 0 .

Therefore, up to a rigid motion, we have parametrized a relative neighborhood ofx0 via the maps ϕ and ψ. By a compactness argument, the whole boundary ∂A1

can be parametrized by a finite number K ∈ N of C1,α-charts (ρi, Ci), where Ci isopen, such that diam (Ci) ≈ R2, for an absolute constant R2 > 0 depending on thegeometry of A1. The following curvature bounds are also satisfied:

(5.37) ‖∂A1‖C1,α ≈∑

i

‖ρi‖C1,α + ‖(ρi)−1‖C1,α ≤M ,

for some M ≥ 1. Moreover we have

ρi(Ci) ⊂ Q+1 and ρi(Ci ∩ ∂A1) ⊂ Q(0, 1) ∩ xn = 0 = Γ1 ,

and

(5.38) det Dρi = det Dρ−1i = 1 , ρ−1

i ≡ ψi : Q1 → Rn .

Boundary comparison estimates. As in (5.1) we define

(5.39) F (x, z) :=∫

B1

F (x+Rx, (u)AR , z)φ(x) dx ,

and obviously

(5.40) (u)AR:= −

∫

AR

u(x) dx .


Finally we define the new functional

(5.41) Fb[w] :=∫

AR

F (x,Dw) dx .

With this definition F (·) satisfies (5.2)-(5.3). Therefore we can apply Proposition8.6 below to the Euler-Lagrange system div Fz(x,Dv) = 0 in AR, obtaining

Theorem 5.1. Let v ∈ u+W 1,p0 (AR,RN ) be the minimizer of the functional Fb[·]

in its Dirichlet class u+W 1,p0 (AR,RN ). Then there exists a constant c, depending

only on n,N, p, L/ν, ‖∂A1‖C1,1 , σ, but otherwise independent of the boundary datumu, the function F (·) and the radius R, such that

(5.42)∫

AR

|Dv|q dx ≤ c∫

AR

(1 + |Du|q) dx .

Before going on we shall provide a suitable bound on the constants appearing inthe Poincare’s inequalities relative to the domain AR. Whenever w ∈W 1,t

0 (AR,RN )and t ≥ 1 then it holds that

(5.43)∫

AR

|w|t dx ≤ c(n, t)Rt

∫

AR

|Dw|t dx ,

and, whenever w ∈W 1,t(AR,RN ) it holds that

(5.44)∫

AR

|w − (w)AR|t dx ≤ c(n, t)Rt

∫

AR

|Dw|t dx .

Both inequalities can be obtained by using their analog on A1, and then scalingback by (5.33). We are now in position to state the following boundary analog ofLemma 5.1.

Lemma 5.3. Let u ∈ W 1,q(Q+1 ,RN ) be a solution to the problem (5.28), where

F (·) satisfies the assumptions (1.30)-(1.32), and where q has been defined in (4.6).Let v ∈ u+W 1,p

0 (AR,RN ) be the unique solution to

minw

∫

AR

F (x,Dw) dx w ∈ u+W 1,p0 (AR,RN ) ,

where F (·) has been defined in (5.39), and assume that (5.5) holds. Then thereexists a positive constant c depending only on n,N, p, L/ν, σ, such that

(5.45)∫

B+R

|V (Du)− V (Dv)|2 dx ≤ c∫

B+3R

(1 + |Du|q + gns) dxRσv

holds, where σv has been defined in (5.7).

Proof. The proof essentially follows the one of Lemma 5.1, where up to a certainstage the domain B must be replaced by AR, and, accordingly (u)B by (u)AR . Themain modifications are the following: estimate (5.9) must be replaced by (5.42),moreover, starting by the last estimate in (5.15) the integral

∫2Bgns dx must be

replaced by∫

B+3Rgns dx everywhere; this comes from the estimate

∫

B1

∫

AR

(g(x) + g(x+Rx))ns dxφ(x)dx ≤ 2ns

∫

B1

∫

B3R

g(x)ns dxφ(x)dx

= 2ns

∫

B3R

gns dx = 2ns

∫

B+3R

gns+1 dx ,

since g(·), initially defined only on Q+1 , has been defined on the whole Q1 by even

reflection. The next modification consists of using Poincare’s inequalities (5.44)


and (5.43) to estimate∫

AR

R−(2γ−p+σv)|u− (u)AR |2γ−p+σv dx ≤ c∫

AR

|Du|2γ−p+σv dx

and ∫

AR

R−τ |u− v|τ dx ≤ c∫

AR

|Du−Dv|τ dxrespectively. Proceeding similarly to the proof of Lemma 5.1 we arrive at thefollowing analog of (5.22)∫

AR

|V (Du)− V (Dv)|2 ≤ cε∫

AR


+c(ε)Rσv

∫

AR


∫

B+3R

gns dx ,

valid for every ε ∈ (0, 1), where c depends only on n,N, p, L/ν. Choosing ε = (2c)−1

in the last estimate, and finally using (5.42) in the resulting inequality yields∫

AR

|V (Du)− V (Dv)|2 dx ≤ c∫

AR

(1 + |Du|q) dxRσv + c

∫

B+3R

gns dxRσv ,

and (5.45) follows taking (5.34) into account. ¤We finally give the boundary version of Lemma 5.2. The statement, whose proof

at this point can be easily obtained combining the proofs of Lemmas 5.2 and 5.3and that we leave to the reader, is

Lemma 5.4. Let u ∈ W 1,qloc (Q+

1 ,RN ) be a solution to the problem (5.28), withF (·) ≡ f(·) + h(·) as in (5.23), where f(·) satisfies (1.41), while h(·) satisfies(1.42), and q has been defined in (4.6). Let v ∈ u + W 1,p

0 (AR,RN ) be the uniquesolution to

minw

∫

AR

f(x,Dw) dx w ∈ u+W 1,p0 (AR,RN ) ,

and assume that (5.5) holds. Then there exists a positive constant c depending onlyon n,N, p, L/ν, g(·), such that

(5.46)∫

B+R

|V (Du)− V (Dv)|2 dx ≤ c∫

B+3R

(1 + |Du|q + |u|q + gns) dxRσ0 ,

holds, where σ0 has been defined in (5.26).

It is here implicit that f(·), h(·) have been extended by even reflection as in(5.30), and then f is the one introduced in (5.24).

5.3. Comparison estimates for systems. Here we shall give comparison esti-mates similar to those in the previous two sections, but for solutions to non-linearelliptic systems as in (1.6). We shall follow the same path adopted in the caseof minimizers: we shall first give interior estimates, and then the boundary ver-sions. In both cases we shall take advantage of the constructions already usedfor minimizers and we shall keep the notation introduced in Sections 5.1-5.2. Weshall start considering a vector field a : Ω × RN × RNn → RNn satisfying (1.44)-(1.45) and (1.31) in Ω × RN × RNn as far as the interior estimates are concerned.When switching to the boundary estimates the vector field a(·) will be defined onQ+

1 × RN × RNn, as for the case of functionals.For the interior estimate we fix an interior ball B such that 4B ⊂ Ω as in Section

5.1. Then we define

(5.47) a(x, z) :=∫

B1

a(x+Rx, (u)B, z)φ(x) dx .


The vector field a(·) satisfies

(5.48)

|a(x, z)|+ |az(x, z)|(1 + |z|2) 12 ≤ L(1 + |z|2) p−1

2

ν(1 + |z|2) p−22 |λ|2 ≤ 〈az(x, z)λ, λ〉

|a(x1, z)− a(x2, z)| ≤ L|x1 − x2|σ(1 + |z|2) p−12 ,

with the same notation and meaning of (1.44)-(1.45).

Lemma 5.5. Let u ∈ W 1,qloc (Ω,RN ) be a local weak solution to (1.6)1 under the

assumptions (1.44)-(1.45) and (1.31), and where q has been defined in (4.6). Letv ∈ u+W 1,p

0 (B,RN ) be the unique solution to

div a(x,Dv) = 0 in Bv = u on ∂B ,

where a(·) defined in (5.47). Then there exists a positive constant c depending onlyon n,N, p, L/ν, σ, such that

(5.49)∫

B

|V (Du)− V (Dv)|2 dx ≤ c∫

2B

(1 + |Du|q + gns) dxRσs ,

where

(5.50) σs := min2α, q + p− 2γ .Proof. The monotonicity of the vector field a(·) implied by (5.48)2 gives

(5.51)1c

∫

B

|V (Du)− V (Dv)|2 dx ≤∫

B

〈a(x,Du)− a(x,Dv), Du−Dv〉 dx .

Using that v solves div a(x,Dv) = 0 and that u solves div a(x, u,Du) = 0 we have∫

B

〈a(x,Dv), Du−Dv〉 dx = 0 =∫

B

〈a(x, u,Du), Du−Dv〉 dx

which in combination with (5.51) gives1c

∫

B

|V (Du)− V (Dv)|2 dx ≤∫

B

〈a(x,Du)− a(x, u,Du), Du−Dv〉 dx:= (I) .(5.52)

We now estimate (I); using definition (5.47) and the assumptions (1.44), we have

(I) =∫

B1

∫

B

〈a(x+Rx, (u)B, Du)− a(x, u(x), Du), Du−Dv〉 dxφ(x) dx

=∫

B1

∫

B

〈a(x+Rx, (u)B, Du)− a(x, (u)B, Du), Du−Dv〉 dxφ(x) dx

+∫

B

〈a(x, (u)B, Du)− a(x, u(x), Du), Du−Dv〉 dx

≤ cRα

∫

B1

∫

B

(g(x+Rx) + g(x))(1 + |Du|2) p−12 |Du−Dv| dxφ(x) dx

+c∫

B

ωα(|u− (u)B|)(1 + |Du|2) p−12 |Du−Dv| dx

=: (II) + (III) .

In turn we have, using Young’s inequality with ε ∈ (0, 1), and Lemma 2.1 as for(5.15)

(II)

≤ cRα

∫

B1

∫

B

(g(x+Rx) + g(x))(1 + |Du|) p2 (1 + |Du|) p

2−1|Du−Dv| dxφ(x) dx


≤ ε∫

B


+c(ε)∫

B1

∫

B

(g(x+Rx) + g(x))2(1 + |Du|)p dxR2α

≤ ε∫

B

|V (Du)− V (Dv)|2 dx+ c(ε)∫

2B

gns dxR2α

+c(ε)∫

B

(1 + |Du|)q dxR2α ,

where of course we have again used (5.16) and (5.17). Again without loss of gener-ality we may assume that γ > p/2, therefore

(III) ≤∫

B

ωα(|u− (u)B|)(1 + |Du|)γ− p2 (1 + |Du|) p

2−1|Du−Dv| dx

≤ ε

∫

B

|V (Du)− V (Dv)|2 dx+ c(ε)∫

B

[ωα(|u− (u)B|)]2(1 + |Du|)2γ−p dx ,

and in turn we have, exactly as in (5.18),∫

B

[ωα(|u− (u)B|)]2(1 + |Du|)2γ−p dx ≤ cRσs

∫

B

(1 + |Du|)q dx .

Combining the last estimate with the preceding one for (III), and in turn combiningthe resulting estimate with the ones found for (I), (II) and finally with (5.52), wefind, for c ≡ c(n,N, p, L/ν), that∫

B

|V (Du)− V (Dv)|2 ≤ cε

∫

B


+c(ε)Rσs

∫

B

(1 + |Du|)q dx+ c(ε)Rσs

∫

2B

gns dx ,

and (5.49) follows choosing ε = (2c)−1. ¤

We now give the boundary version of the previous lemma. Here we shall keepthe notation concerning boundary problems introduced in Section 5.2. We shallconcentrate our attention on the following (partial) boundary-value problem:

(5.53)

div a(x, u,Du) = 0 in Q+1

u = 0 on Γ1 ,

under the assumptions (1.44)-(1.45) and (1.31), and relative notation, with Ω ≡ Q+1 .

In particular we shall assume that a(·) satisfies

(5.54) |a(x1, y, z)− a(x2, y, z)| ≤ (g(x1) + g(x2))|x1 − x2|α(1 + |z|2) p−12

for almost every (x1, x2) ∈ Q+1 ×Q+

1 , and every choice (y, z) ∈ RN × RNn.As in (5.30) we extend a(·) by even reflection, getting a new vector field defined

on Q1 × RN × RNn → RNn, and still denoted by a(·), that is

(5.55) a(x, y, z) :=

a(x, y, z) x ∈ Q+1

a(i(x), y, z) x ∈ Q−1 .

After having also extended g(·) as in (5.31), it is easy to verify that the extendedvector field a(·) still satisfies (1.44)-(1.45) and (1.31) with Ω ≡ Q1. As for thevariational case, we select a ball BR centered on xn = 0 and such that B4R ⊂ Q1,and we consider the related domain AR as in (5.34). We define the comparisonvector field a(·) as

(5.56) a(x, z) :=∫

B1

a(x+Rx, (u)AR , z)φ(x) dx ,


where (u)ARis as in (5.40). The boundary version of Lemma 5.5 is now

Lemma 5.6. Let u ∈ W 1,q(Q+1 ,RN ) be a solution to (5.53), under the assump-

tions (1.44)-(1.45) and (1.31), where q has been defined in (4.6). Let v ∈ u +W 1,p

0 (AR,RN ) be the unique solution to

(5.57)

div a(x,Dv) = 0 in AR

v = u on ∂AR ,

where a(·) has been in defined in (5.56). Then there exists a positive constant cdepending only on n,N, p, L/ν, σ, such that∫

B+R

|V (Du)− V (Dv)|2 dx ≤ c∫

B+3R

(1 + |Du|q + gns) dxRσs ,

where σs has been defined in (5.50).

The proof of the previous lemma follows verbatim the one of Lemma 5.5, replac-ing B by AR, and taking into account the proof of Lemma 5.3.

5.4. Additional regularity properties. Here we state an additional regularityproperty of the vector field defined in (5.47); this observation is essentially theone allowing us to deal with the case of fractional Sobolev coefficients (1.30)4 and(1.44)3.

Lemma 5.7. Let a(·) be the vector field defined in (5.47), then it holds that

(5.58) |ax(x, z)| ≤ c(L)Rα−1(g(x) + gR(x))(1 + |z|2) p−12 ,

whenever x ∈ B and z ∈ RNn, where

(5.59) gR(x) :=∫

B1

g(x+Rx) dx .

In addition let a(·) be the functional defined in (5.56), then it also holds

(5.60) |Fzx(x, z)| ≤ c(L)Rα−1(g(x) + gR(x))(1 + |z|2) p−12 .

Inequalities (5.58)-(5.60) also hold for a(·) defined in (5.56), and for Fz(·) definedin (5.39), respectively. For the boundary cases (5.56) and (5.39) by g(·) we meanthe function extended as in (5.31).

Proof. We first give the proof of (5.58). Recall that φ(·) is a symmetric mollifierand therefore ∫

B1

Dφ(x) dx = 0 .

Hence (5.58) follows by (1.44)3 and the following estimations:

|ax(x, z)| =

∣∣∣∣∣−1

Rn+1

∫

BR(x)

⟨a(x, (u)B, z), Dφ

(x− xR

)⟩dx

∣∣∣∣∣

=∣∣∣∣1R

∫

B1

〈a(x+Rx, (u)B, z), Dφ(x)〉 dx∣∣∣∣

=∣∣∣∣1R

∫

B1

〈[a(x+Rx, (u)B, z)− a(x, (u)B, z)], Dφ(x)〉 dx∣∣∣∣

≤ cRα−1 supB1

|Dφ|∫

B1

(g(x+Rx) + g(x)) dx (1 + |z|2) p−12

≤ cRα−1(g(x) + gR(x))) (1 + |z|2) p−12 .

In the same way (5.60) follows using (1.30)4 and the very definition of F [·] in(5.1). ¤


6. Conditional fractional estimates

Here we prove fractional Sobolev spaces estimates for solutions to partial Dirich-let problems i.e. defined in an upper cube, but with the Dirichlet condition imposedon lower side only. The point is that we shall assume that solutions satisfy addi-tional integrability properties i.e. u ∈W 1,q for some q > p.

6.1. Functionals. We consider a solution u ∈W 1,p(Q+1 ,RN ) to the Dirichlet vari-

ational problem (5.28). We shall assume that

(6.1) u ∈W 1,q(Q+d ,R

N ) for every d ∈ (0, 1) ,

and q has been defined in (4.6). The functions F (·) and g(·) are defined on Q1 ×RN × RNn and Q1, respectively, via the even extensions in (5.30)-(5.31). Beforegoing on we also extend u on the whole cube Q+

1 by even reflection: again denotingx ≡ (x′, xn), and i : Rn 3 (x′, xn) 7→ (x′,−xn) ∈ Rn, we let

(6.2) u(x) :=

u(x) x ∈ Q+1

u(i(x)) x ∈ Q−1 .We also recall the definition of the regularized function gR(·), defined in (5.59) forR > 0. The result is now

Proposition 6.1. Let u ∈ W 1,p(Q+1 ,RN ) be a solution to (5.28), under the as-

sumptions (1.30)-(1.32) with Ω ≡ Q+1 . Assume in addition that u satisfies (6.1)

and that (5.5) holds, that is

(6.3)τγ

τ − β ≤ q where 2 ≤ τ ≤ p .Then

(6.4) V (Du) ∈W θ,2(Q+d ,R

N ) and Du ∈W 2θp ,p(Q+

d ,RN )

for every θ < σv/2 and 0 < d < 1, where σv ≡ σv(τ) has been defined in (5.7).Moreover, for every 0 < d < 1 there exists a constant c, depending on the datan,N, p, ν, L, d, σ, such that

(6.5) [V (Du)]2θ,2;Q+

d

+ [Du]p2θ/p,p;Q+

d

≤ c∫

Q+1

(1 + |Du|q + gns) dx .

For the proof, let us define

(6.6) γ(t) :=σv/2

σv/2 + 1− t , for every t ∈ [0, σv/2 + 1) .

The proof of Proposition 6.1 is in turn based on the following:

Lemma 6.1. Under the assumptions of Proposition 6.1 assume that

(6.7) V (Du) ∈W t,2(Q+d ,R

Nn) , for some t ∈ [0, σv/2) ,

and for every d ∈ (0, 1), and that whenever 0 < d1 < d2 < 1 there exists c1depending on d2 − d1, such that

(6.8) [V (Du)]2t,2;Q+

d1

≤ c1∫

Q+d2

(1 + |Du|q + gns) dx .

Then

(6.9) V (Du) ∈W t,2loc (Q

+d ,R

Nn) , for every t ∈ [0, γ(t)) ,

where γ(·) is defined in (6.6), and moreover, with 0 < d1 < d2 < 1 being fixed, thereexists a new constant c2 depending only on n,N, p, L/ν, σ, d2 − d1, c1, such that

(6.10) [V (Du)]2t,2;Q+

d1

≤ c2∫

Q+d2

(1 + |Du|q + gns) dx .


Finally, for every s ∈ 1, . . . , n, and for h ≤ d2 − d1, it holds that

(6.11) suph

∫

Q+d1

|τs,hV (Du)|2|h|2γ(t)

dx ≤ c∫

Q+d2

(1 + |Du|q + gns) dx .

In the case s = n the last inequality holds under the additional condition that h > 0.

Proof. Step 1: Estimates on balls. Without loss of generality, in order to avoidto pass to smaller cubes too many times, we shall assume that (6.1) holds withd = 1. We next fix a notation that we shall keep for the rest of the paper. Letus take B b Ω, a ball of radius R; we denote by Qinn ≡ Qinn(B) the largest cube,concentric to B and with sides parallel to the coordinate axes, contained in B;clearly |B| ≈ |Qinn|. The cube Qinn(B) will be called the inner cube of B, andconversely B will be called the outer ball of Qinn(B). We also denote the enlargedball as B ≡ 16B. We have the following chain of inclusions:

(6.12) Qinn(B) ⊂ B b 2B b 4B ⊂ B = 16B .

Now we fix 0 < d1 < d2 < 1, and then take β0 ∈ (0, 1) to be specified later, andlet h ∈ R be a real number satisfying

(6.13) 0 < |h| ≤(d2 − d1

10000n4

) 1β0

=: d <d2 − d2

10000n4.

We start by considering a ball of radius |h|β0 , and such that

(6.14) x0 ∈ Q+d1

and B ≡ B(h) = B(x0, |h|β0) b B ⊂ Q+d2.

Remark 6.1. Denoting x0 ≡ ((x0)i)i≤n, we observe that the condition

(6.15) (x0)n ≥ 16|h|β0 ,

together with (6.13), ensures the validity of the last inclusion in (6.14) - moreprecisely (6.15) ensures that B does not touch the lower boundary xn = 0, whilesince x0 ∈ Q+

d1, (6.13) ensures that B does not touch the lateral boundary of B.

Let us first define v ∈ u + W 1,p0 (4B) as the unique solution to the following

Dirichlet variational problem:

(6.16) minw

∫

4B

F (x,Dw) dx w ∈ u+W 1,p0 (4B,RN ) ,

where F (·) has been defined in (5.1) with B ≡ 4B. Therefore v satisfies the followingEuler-Lagrange system:

(6.17) div Fz(x,Dv) = 0 ,

in 4B. Moreover, applying (5.9) we have

(6.18)∫

4B

|Dv|q dx ≤ c∫

4B

(1 + |Du|q) dx .

We can therefore apply Theorem 4.4 to the system (6.17); the assumptions aresatisfied with

(6.19) γ(·) = |h|β0(α−1)[g(·) + g|h|β0 (·)] ,by virtue of Lemma 5.7, (5.60). Moreover, notice that assumption (4.13) withA ≡ 4B is satisfied since g(·) ∈ Lns(Q1) and (6.18) holds; see (6.21) later on.Therefore we conclude with∫

2B

|D(V (Dv))|2 dx ≤ c|h|−2β0

∫

4B

|V (Dv)− V |2 dx

+c∫

4B

γ2(1 + |Dv|)p dx ,(6.20)


whenever V ∈ RNn; we remark that in the last estimate, as also in the followingones, the constant c will always be independent of h and V . It will now appearin a clear way the central role played by the precise form of estimate (4.14). Topreliminarily estimate the last integral in (6.20) we use (4.9), Young’s inequalityand (6.18)

(6.21)∫

4B

γ2(1 + |Dv|)p dx ≤ c|h|2β0(α−1)

∫

B

(1 + |Du|q + gns) dx ,

where we also used the definition of g|h|β0 (·) in (5.59) with R = |h|β0 - comparewith (5.16). Combining the last two estimates we then have

∫

2B

|D(V (Dv))|2 dx ≤ c|h|−2β0

∫

4B

|V (Du)− V |2 dx

+c|h|2β0(α−1)

∫

B

(1 + |Du|q + gns) dx+ c|h|−2β0

∫

4B

|V (Du)− V (Dv)|2 dx .(6.22)

Now, fix s ∈ 1, . . . , n; due to (6.13), and since |h| ≤ |h|β0 , we have that B+hes ⊂2B ⊂ B, and a standard result on difference quotients yield

∫

B

|τs,hV (Dv)|2 dx ≤ |h|2∫

2B

|DsV (Dv)|2 dx

for each s ∈ 1, . . . , n. Combining this last estimate with (6.22) we have∫

B

|τs,hV (Dv)|2 dx ≤ c|h|2(1−β0)

∫

4B

|V (Du)− V |2 dx

+c|h|2β0α

∫

B

(1 + |Du|q + gns) dx+ c|h|2(1−β0)

∫

4B

|V (Du)− V (Dv)|2 dx .

The estimate for the last integral in the previous inequality is now provided byLemma 5.1 applied with B ≡ 4B, and therefore we conclude with

∫

B

|τs,hV (Dv)|2 dx ≤ c|h|2(1−β0)

∫

B

|V (Du)− V |2 dx

+c|h|β0σv

∫

B

(1 + |Du|q + gns) dx ,(6.23)

where we used (6.13) to estimate |h|2(1−β0) ≤ 1, and σv ≤ 2α to estimate |h|2β0α ≤|h|β0σv . Note that the application of Lemma 5.1 is possible since (5.5) holds byassumption.

In turn, using again the fact that B + hes ⊂ 2B ⊂ B we get∫

B

|τs,hV (Du)|2 dx ≤ c

∫

B

|τs,hV (Dv)|2 dx

+c∫

B

|V (Du(x+ hes))− V (Dv(x+ hes))|2 dx

+c∫

B

|V (Dv)− V (Du)|2 dx

≤ c

∫

B

|τs,hV (Dv)|2 dx+ c

∫

4B

|V (Du)− V (Dv)|2 dx .(6.24)

Combining the last estimate again with Lemma 5.1 and (6.23) yields∫

B

|τs,hV (Du)|2 dx ≤ c|h|2(1−β0)

∫

B

|V (Du)− V |2 dx

+c|h|β0σv

∫

B

(1 + |Du|q + gns) dx .(6.25)


We recall that the last inequality holds whenever V ∈ RNn and that the constantc is still independent of h and V .

We now wish to obtain a boundary version of (6.25). More precisely, we shallconsider an upper ball centered on Γd1 ⊂ xn = 0. The proof remains essentiallythe same, but we have to use the comparison results of Section 5.2, rather thanthose of Section 5.1; we have to keep in mind that in the case s = n - this is thenon-tangential direction - we have h > 0. Let us give some details - although themost part of the arguments are parallel to those for estimate (6.25).

We consider a ball B ≡ B(h) ≡ B16√

n|h|β0 centered on Γd1 , with h still satisfying(6.13), and the relative outer ball B. Accordingly, we determine the correspondingupper ball B+

16√

n|h|β0, and the relative outer ball B+. Note that by virtue of (6.13),

although we are taking a larger radius, that is 16√n|h|β0 instead of |h|β0 , we still

have that

(6.26) B+ ⊂ Q+d2.

Next we determine the intermediate C2-regular set A ≡ A(h) as in (5.34)

(6.27) (4B)+ ⊂ A(h) ⊂ (8B)+ .

We define v ∈ u+W 1,p0 (A(h),RN ) be the unique solution to

(6.28) minw

∫

A(h)

F (x,Dw) dx w ∈ u+W 1,p0 (A(h),RN ) ,

where F (·) is defined in (5.39) - note that with the notation of Section 5.2 herewe are taking, up to a translation, AR ≡ A(h), and R = 64

√n|h|β0 . Therefore

Theorem 5.1 implies that∫

(4B)+|Dv|q dx ≤

∫

A(h)

|Dv|q dx ≤ c

∫

A(h)

(1 + |Du|q) dx

≤ c

∫

B+(1 + |Du|q) dx .(6.29)

We stress that it is crucial here that in (6.29) the constant c is independent ofh. Finally, the map v satisfies the Euler-Lagrange system (6.17) in A(h), withF (·) being defined in (5.39). We have now all the tools for getting the desiredboundary version of estimate (6.25); we use Theorem 4.3 (instead of Theorem 4.4)with A ≡ A(h) to get (6.20) with 2B and 4B replaced by (2B)+ and (4B)+,respectively. Then we use Lemma 5.3 instead of Lemma 5.1, and (6.29) insteadof (6.18), and proceeding as after (6.18), and using (6.27) repeatedly, we get theboundary analog of (6.25), that is

∫

B+|τs,hV (Du)|2 dx ≤ c|h|2(1−β0)

∫

B+|V (Du)− V |2 dx

+c|h|2β0(σv/2)

∫

B+(1 + |Du|q + gns) dx(6.30)

which holds whenever V ∈ RNn, for a constant c which is independent of h and V .Observe that since g(·) has been also extended to the whole Q1 by even reflection,then Theorem 4.3 still works with the choice (6.19) - compare with the statementof Lemma 5.7.

Step 2: Covering construction. In the following h will go on denoting a realnumber satisfying (6.13). We now consider Qd1 as with d1 < d2 < d, and coverit with a family of closed cubes Qii, with sides parallel to the coordinate axes,such that the family of open cubes Qii is disjoint, and such that the sidelength ofevery such cube is comparable to |h|β0 via a constant which is independent of the


index i; moreover, every such cube will touch Qd1 , and this means that all cubesare needed to cover Qd1 . As a first corollary, since the number of cubes is finiteand their boundaries are negligible, we have

(6.31)∣∣∣Q+

d1\

⋃(Qi ∩Q+

d1)∣∣∣ = 0 .

The precise construction needed is now as follows: the cubes meeting the hyperplanexn = 0 will be slightly larger than the others, and all of them will share thesame sidelength, that will be exactly 32|h|β0 ; they will be therefore equal up to atranslation. Moreover they will be chosen such that |Q+

i | = |Q−i | so that they aredivided in exactly two parts by the hyperplane xn = 0. The remaining ones,those not touching xn = 0, will have sidelength equal to 2|h|β0/

√n. This ends

the initial construction of the cubes.In the next step we realize such cubes as the inner cubes of a certain family of

balls Bii, which are therefore uniquely determined by such condition, i.e.

(6.32) Qi ≡ Qinn(Bi) .

We have therefore two kinds of balls. Those relative to the cubes which do nottouch xn = 0 have radius |h|β0 , and are of the type considered in (6.14). Indeed,call x0 ≡ ((x0)i)i≤n the center of such a cube, Q, of course x0 is also the centerof the relative outer ball, say B (it is Q ≡ Qinn(B)); since Q cannot touch thosecubes Qi cut by the hyperplane, and since these have sidelengths equal to 32|h|β0 ,we conclude that (6.15) is satisfied by x0, and therefore (6.14) holds for the ballsrelative to such cubes. Accordingly, on such balls we are later going to applyestimate (6.25).

Then there are those balls which are the outer balls of the larger cubes i.e. thosecubes cut by xn = 0 in two equal pieces; these have radius equal to 16

√n|h|β0 . In

turn such balls determine upper balls B+i , and since such balls are still touching

Qd1 , due to the size restriction in (6.13), we still have that B+i ⊂ Q+

d2- compare

with (6.26). On such upper balls B+i we are later going to apply estimate (6.30).

Finally, concerning the whole family of all the balls Bi, determined in connectionto Bi according to (6.32) and Bi = 16Bi, they are still contained in Qd2

(6.33) Bi ⊂ Qd2

and, more importantly, they enjoy the finite intersection property: there is a uni-versal constant c ≡ c(n) such that every ball Bi meets at most c(n) other similarballs Bii. This is a direct consequence of their definition, and of the fact thatthe initially considered family Qii was a disjoint lattice of cubes with sidelengthsequal to 2|h|β0/

√n or 32|h|β0 and sides parallel to the coordinate axes. Finally, by

(6.31) and (6.32), it follows that

(6.34)∣∣∣Q+

d1\

⋃(Bi ∩Q+

d1)∣∣∣ = 0 .

The construction of the covering is now complete.Step 3: Fractional estimates. We finally prove the fractional differentiability

of V (Du) and Du. We start with the case t = 0. We first choose β0 ∈ (0, 1) inorder to have

(6.35) 2(1− β0) = β0σv ⇐⇒ β0 =1

σv/2 + 1.

Note that β0 < 1, obviously. Now let us define the measure

(6.36) µ(A) :=∫

A

(1 + |Du|q + gns) dx A ⊂ Q1 ,


and recall that u has been extended on the whole Q1. According to the constructionof Step 2 we shall use estimate (6.25) on the balls Bi, which are not touchingxn = 0, and estimate (6.30) on the upper balls B+

i ; here we are adopting theimplicit convention/notation according to what the only half balls considered arethose relative to the balls touching xn = 0, while all the others do not touchxn = 0 by construction. In (6.25)-(6.30) we take V = 0, independently of theball considered. By (6.34) and (6.35) we then have∫

Q+d1

|τs,hV (Du)|2 dx ≤∑

i

∫

Bi

|τs,hV (Du)|2 dx+∑

i

∫

B+i

|τs,hV (Du)|2 dx

≤ c|h|β0σv

[∑

i

∫

Bi

|V (Du)|2 dx+∑

i

∫

B+i

|V (Du)|2 dx]

+c|h|β0σv

[∑

i

µ(Bi) +∑

i

µ(B+i )

]

≤ c|h|β0σv

[∑

i

µ(Bi) +∑

i

µ(B+i )

].

Note that in the last line we have estimated

(6.37) |V (Du)|2 ≤ c(1 + |Du|p) ≤ c(1 + |Du|q) .Taking into account that the family Bi has the finite intersection property - seeStep 2 - and that (6.33) holds, we conclude with

∫

Q+d1

|τs,hV (Du)|2 dx ≤ c|h| σvσv/2+1µ(Qd2) .

Here we used (6.35) again. Finally, using the fact that u is symmetric with respectto the hyperplane xn = 0, and recalling (6.6), we have

(6.38)∫

Q+d1

|τs,hV (Du)|2 dx ≤ c|h|2γ(0)µ(Q+d2

) .

The last estimate works for every s ∈ 1, . . . , n, and whenever h satisfies (6.13),with h > 0 when s = n. Therefore (6.11) immediately follows when t = 0 for hsatisfying (6.13); in the case d ≤ |h| ≤ d2 − d1 inequality (6.11) follows enlargingthe constant c in (6.38) of a factor comparable to 1/(d2 − d1)2, and using (6.37)again.

Applying now Lemma 3.1, and slightly changing d1 - taking for instance (d1 +d2)/2, we recall that here d1 and d2 can be arbitrarily chosen - we get (6.9) withthe estimate (6.10). The proof of the lemma is complete in the case t = 0.

For the case t > 0 we need to refine the previous argument. Let us take β0 ∈ (0, 1)such that

(6.39) 2(1− β0 + t) = β0σv ⇐⇒ β0 =1

σv/2 + 1− t .

Note that β0 < 1, since we are assuming that t < σv/2. This time we use the setfunction(6.40)

λ(A) := µ(A) +∫

A

∫

A

|V (Du(x))− V (Du(x))|2|x− x|n+2t

dx dx = µ(A) + [V (Du)]22,t;A

defined on Borel subsets A ⊂ Q1, where µ(·) has been already defined in (6.36). Werecall that the current assumption is (6.7); recalling that u has been symmetricallyextended to the whole Q1 in (6.2), and using Proposition 3.2, we have that V (Du) ∈


W t,2(Qd,RNn) for every d ∈ (0, 1). Therefore the definition of λ(·) makes sense andthe function is always finite. Now we use (6.25) with B ≡ Bi and V = (V (Du))Bi

.Moreover, we use the fractional Poincare’s inequality (3.6) to get

|h|2(1−β0)

∫

Bi

|V (Du)− (V (Du))Bi|2 dx ≤ c|h|2(1−β0+t)[V (Du)]2

2,t;Bi,

and therefore, taking into account (6.39) and (6.40), we conclude with

(6.41)∫

Bi

|τs,hV (Du)|2 dx ≤ c|h| σvσv/2+1−tλ(Bi) .

As for the upper balls B+i we proceed similarly, applying (6.30) on B+

i . Accord-ingly, we choose V = (V (Du))B+

iand use Poincare’s inequality on half balls - see

Proposition 3.3 - to get

|h|2(1−β0)

∫

B+i

|V (Du)− (V (Du))B+i|2 dx ≤ c|h|2(1−β0+t)[V (Du)]2

2,t;B+i

≤ c|h|2(1−β0+t)[V (Du)]22,t;Bi

,(6.42)

and similarly to (6.41), the last inequality and (6.30) give

(6.43)∫

B+i

|τs,hV (Du)|2 dx ≤ c|h| σvσv/2+1−tλ(Bi) .

We now sum up inequalities (6.41) and (6.43) over the family Bi, B+i . To facili-

tate this we premise the observation that although failing to be a measure by thepresence of the term [V (Du)]22,t;·, the set function λ(·) is nevertheless easily seen tobe a countably super-additive set function, meaning that

(6.44)∑

λ(Ai) ≤ λ (∪Ai)

holds whenever Aii is a countable family of mutually disjoint subsets. Therefore,summing again over the family Bi, and recalling again that such a family has thefinite intersection property to use (6.44), by (6.34) we have

∫

Q+d1

|τs,hV (Du)|2 dx ≤∑

i

∫

Bi

|τs,hV (Du)|2 dx+∑

i

∫

B+i

|τs,hV (Du)|2 dx

≤ c|h| σvσv/2+1−t

[∑

i

λ(Bi) +∑

i

λ(B+i )

]

≤ c|h| σvσv/2+1−tλ(Qd2) ,

and where we have used (6.39). Taking again into account the definition of γ(·) in(6.6), and using (6.39), we arrive at

(6.45)∫

Q+d1

|τs,hV (Du)|2 dx ≤ c|h|2γ(t)λ(Qd2) ≤ c|h|2γ(t)λ(Q+d2

) .

Now using again Lemma 3.1, and taking into account that d1 < d2 are arbitrary,after slightly changing the values of d1, d2, (6.45) yields

(6.46) [V (Du)]2t,2;Q+

d1

≤ c∫

Q+d2

(1 + |Du|q + gns) dx+ c[V (Du)]2t,2;Q+

d2

,

whenever t < γ(t). On the other hand, since t < γ(t) as long as t < σv/2, then,after again slightly changing the values of d1, d2, we can use assumption (6.8) toestimate the last term in (6.46), thereby finally getting (6.10). In the same way weproceed to estimate the right hand side in (6.45), which yields (6.11) for h satisfying


(6.13). The validity of (6.11) for |h| ≤ d2 − d1 again follows enlarging the constantby a factor 1/(d2 − d1)2, and the proof of the whole lemma is complete. ¤Proof of Proposition 6.1. The proof is a now a consequence of an iteration argumentbased upon Lemma 6.1. One starts applying Lemma 6.1 with θ = 0, and improveson the fractional differentiability of V (Du) via an iterated application of Lemma6.1. More precisely V (Dv) ∈ W θn−ε,2

loc (Ω,RNn), for every ε > 0 and every n ∈ N,where the sequence θnn is inductively defined by

θn+1 := γ(θn) θ0 = 0 .

Note that θn monotonically increases to the fixed point σv/2 = γ(σv/2), andthis yields the inclusion (6.4). The a priori estimate (6.5), for the part concerningV (Du), follows iterating the estimates found in Lemma 6.1. Finally, using (6.11)and Lemma 3.2 o ne can obtain the results for Du from those valid for V (Du). Fora detailed exposition we refer to [36, Theorem 4.2] and [43, Lemma 6.3] where asimilar iteration argument has been applied. ¤

We finally state a version of Proposition 6.1 relative to the case considered inTheorem 1.8.

Proposition 6.2. Let u ∈W 1,p(Q+1 ,RN ) be a solution to the problem (5.28), with

F (·) ≡ f(·)+h(·) as in (5.23), where f(·) satisfies (1.41), while h(·) satisfies (1.42),and q has been defined in (4.6). Assume that (6.1) and (6.3) hold. Then (6.4) holdsfor every θ < σ0/2 and 0 < d < 1, where σ0 has been defined in (5.26). Moreover,for every 0 < d < 1 there exists a constant c, depending on n,N, p, ν, L, σ, d, suchthat (6.5) holds with |Du|q replaced by |Du|q + |u|q.

The proof of Proposition 6.2 is completely similar to that of Proposition 6.1,and relies on a similar version of Lemma 6.1 working under the assumptions ofTheorem 1.8, and that the reader will have no problem in formulating him/herself.The only change worth mentioning is that, when considering the comparison mapsv ∈ u + W 1,p

0 (4B,RN ) and v ∈ u + W 1,p0 (A(h),RN ), solving (6.16) and (6.28)

respectively, one considers, instead of F (·), the simpler energy density f(·) definedin (5.24), with B ≡ 4B; note that Lemma 5.7 obviously applies to fz(·). The restof the proof follows verbatim the one for Lemma 6.1, but using Lemmas 5.2 and5.4, instead of Lemmas 5.1 and 5.3, respectively. Note that in such a proof, asa consequence of inequalities (5.25) and (5.46), the definition of the measure µ(·)appearing in (6.36) changes to

µ(A) :=∫

A

(1 + |Du|q + |u|q + gns) dx .

Accordingly, the quantity |Du|q appearing in (6.5), (6.8), (6.10) and (6.11), isreplaced by |Du|q + |u|q.6.2. Systems. We now come to state the corresponding fractional differentiabilityresults for solutions to partially Dirichlet problems as

(6.47)

div a(x, u,Dv) = 0 in Q+1

u = 0 on ∂Γ1 .

The vector field a(·) is assumed to satisfy (1.44)-(1.45) and (1.31) with Ω ≡ Q+1 ,

and we shall extend it by even reflection as in (5.55). Moreover, as usual also g(·)and u are extended as in (5.31) and (6.2), respectively, to the whole Q1. Severalparts of the proof will be similar to those for the case of functionals, and we shallconfine ourselves to sketching the required modifications. A main point is that thedifferentiability exponent σv defined in (5.7) and used in Proposition 6.1 will bealways replaced by the new exponent σs defined in (5.50).


We shall proceed as in Section 6.1, therefore assuming (6.1) with d = 1.

Proposition 6.3. Let u ∈ W 1,p(Q+1 ,RN ) be a solutions to (6.47), where a(·)

satisfies (1.44)-(1.45) and (1.31), and assume that (6.1) holds. Then

V (Du) ∈W θ,2(Q+d ,R

N ) and Du ∈W 2θp ,p(Q+

d ,RN )

holds for every θ < σs/2 and 0 < d < 1, where σs has been defined in (5.50). More-over, for every 0 < d < 1 there exists a constant c, depending on n,N, p, ν, L, σ, d,such that (6.5) holds.

Accordingly, we define

(6.48) γ(t) :=σs/2

σs/2 + 1− t , for every t ∈ [0, σs/2 + 1) ,

and the proof of Proposition 6.3 relies on the following:

Lemma 6.2. Under the assumptions of Proposition 6.1 the same result of Lemma6.1 holds for the solutions u to (6.47) provided σv is replaced by σs defined in (5.50),and the function γ(·) is accordingly the one defined in (6.48).

Proof. The proof follows the one given for Lemma 6.1, and we shall describe theneeded modifications. We consider a ball as in (6.14) and this time the comparisonmap v ∈ u + W 1,p

0 (4B,RN ) is defined as the unique solution to the followingDirichlet problem:

(6.49)

div a(x,Dv) = 0 in 4Bv = u on ∂4B ,

where the new vector field a(·) is the one defined in (5.47) with B ≡ 4B. Moreover(6.18) holds by Proposition 8.6 below, and we can again apply Theorem 4.4 tothe system (6.49)1 with γ(·) as in (6.19) by virtue of Lemma 5.7. Then we mayproceed exactly as in Lemma 6.2, arriving up to (6.25). We then pass to analyzethe situation on upper balls. We again determine the domain A(h) as in (6.27) andwe define v ∈ u+W 1,p

0 (A(h),RN ) to be the unique solution to

(6.50)

div a(x,Dv) = 0 in A(h)v = u on ∂A(h) ,

and this time a(·) has been defined in (5.56) with AR ≡ A(h). Again (6.29) followsthis time by Proposition 8.6, and moreover we can apply Theorem 4.3 to the system(6.50)1, in order to finally get (6.30). Once (6.25) and (6.30) have been established,the rest of the proof proceeds exactly as for Lemma 6.1, Steps 2 and 3. ¤

7. Existence of regular boundary points

Here the plan is the following: We shall first give the proof of Theorems 1.3-1.5.All the other theorems from Sections 1.1 and 1.3 except Theorem 1.8, follow asparticular cases; note that in order to treat functionals as in (1.16) or in (1.39)-(1.40) we have to recall Remark 1.3. We shall then obtain, via arguments similar tothose for Theorem 1.5, the proof of Theorem 1.8. Finally we shall treat the case ofsystems, proving Theorems 1.10 and 1.11. All proofs have large parts in common.

We first start recalling the standard flattening-of-the-boundary procedure allow-ing to reduce the boundary regularity of problems as (1.1) to that of regularityup to Γ1 of problems as (5.28), and that of problems as (1.6) to that of problemsas (5.53). This procedure is standard and for the case of systems is described in[17, Section 5], or [11, Section 8]. We therefore recall the procedure in the case offunctionals also for future convenience. Observe that by means of the new integrand

(x, y, z) 7→ F (x, u0(x) + y,Du0(x) + z) =: F0(x, y, z) ,


we can reduce to the case u0 ≡ 0 in (1.1). Indeed, using that u0 ∈ C1,α(Ω,RN ) it iseasy to see that if F (·) satisfies (1.30)-(1.32) then F0(·) still satisfies (1.30)-(1.32),provided we replace σ by minσ, α, and for a new choice of the constants ν, L, nowdepending on ‖u0‖C1,α . Therefore, from now on we can reduce to a minimizationproblem amongst maps with zero boundary value.

Next we flatten ∂Ω; we follow the terminology of Remark 5.1: we consider afinite number of C1,α-charts (ρi, Ci) covering Ω and parametrizing ∂Ω. Here Ci areopen domains such that ρi(Ci) ⊂ Q+

i where Qi are cubes centered at the originand such that ρi(∂Ci) ⊂ Qi ∩ xn = 0. We concentrate on one of such charts, say(ρi, Ci); without loss of generality we assume that Qi ≡ Q1; moreover, denotingψ = ρ−1

i , we may assume ψ : Q+1 ∪Γ1 → Rn. We define the new integrand - actually

the pull-back of F (·, u0(·) + ·, Du0(·) + ·) via ψ - by

(x, y, z) 7→ F (ψ(x), u0(ψ(x)) + y, (Du0)(ψ(x)) + z(Dψ(x))−1)=: F1(x, y, z) , x ∈ Q+

1 ∪ Γ1 .(7.1)

Keep in mind det Dψ = 1, compare with Remark 5.1. Let us first notice that thefunction g ψ is measurable and belongs to Lns(Q+

1 ). At this point we again havethat if F (·) satisfies (1.30)-(1.32), also F1(·) satisfies the same assumptions, providedwe choose new constants ν, L depending on ‖ψ‖C1,α , and again with σ replaced byminσ, α; from now on we shall therefore assume that σ < α. Assumption (1.30)4will be satisfied by F1(·) using g ψ.

Moreover we have that the map u ψ is a solution to (5.28) with F (·) ≡ F1(·).Therefore, as in the more standard case of systems [17], the study of boundaryregularity of minima is reduced to the special flat case (5.28), and we are reducedto prove that almost every point of Γ1 is a regular point for u ≡ u ψ. In the sameway, it will be sufficient to prove (1.21) for u ≡ u ψ. In turn, denoting by Σb

u

the set of singular boundary points - specifically Σbu = Γ1 \ Ωb

u, with the notationintroduced in (1.10) - this will be achieved by showing

(7.2) dimH(Σbu) < n− 1 .

We now recall and restate the boundary regularity criteria available in the literature- see [6, 39]. We have that

(7.3) Σbu = Σ0

b ∪ Σ1b ,

where

Σ0b :=

x ∈ Γ1 : lim inf

%0−∫

B(x,%)∩Q+1

|Du(x)− (Du)B(x,%)∩Q+1|p dx > 0

or lim sup%0

|(Du)B(x,%)∩Q+1| =∞

,

and

Σ1b :=

x ∈ Γ1 : lim inf

%0−∫

B(x,%)∩Q+1

|u(x)− (u)B(x,%)∩Q+1|p dx > 0

or lim sup%0

|(u)B(x,%)∩Q+1| =∞

.

Now since it trivially holds that u ∈ W 1,p(Q+1 ,Rn) then applying Proposition 3.1

it follows that dimH(Σ1b) ≤ n− p < n− 1. In any case it remains to show that

(7.4) dimH(Σ0b) < n− 1 .


Proof of Theorems 1.3-1.4. We start with Theorem 1.3; for this we appeal to Theo-rem 4.2, applicable as 2 < n ≤ p+2 holds. Since dimH(Ω\(Ωu∪Ou)) ≤ n−p < n−1,by discarding a relatively - with respect to Ω - closed (and hence closed) set of Haus-dorff dimension strictly less than n− 1, we can reduce to the situation in which weestimate those parts of the singular set contained in Ωu ∪ Ou. This means - recallthat Ou is relatively open with respect to ∂Ω - that when flattening the boundaryand passing to the flat problem (5.28), we can assume that u is globally Holder con-tinuous in Q+

1 ∪ Γ1 when proving (7.2), and moreover we may assume the Morreyspace condition Du ∈ Lp,µ1(Q+

1 ∪ Γ1,RNn) with µ1 ∈ (2, n). At this point we canapply Proposition 8.1 below with H ≡ (Du0) ψ ∈ L∞ and h ≡ u0 ψ ∈ C0,α, inorder to deduce that Du ∈ Lq(Q+

d ,RNn) for every d < 1; therefore (6.1) is satisfiedand we want to apply Proposition 6.1. To this aim it remains to check the validityof (6.3) for a suitable number τ ∈ [2, p] we are going to properly choose now. Wenotice that from now on we shall use only the condition n ≤ 2p + 2 assumed inTheorem 1.4, which in particular is implied by the condition n ≤ p + 2 assumedin Theorem 1.3. We first treat the case n ≥ 6, where assumption (1.33) givesβ > 1− 2/n; in this case we take

(7.5) τ =n− 2

2.

It turns out that τ ∈ [2, p], as required in (6.3); indeed τ ≥ 2 holds as we areassuming n ≥ 6, while the low dimensional assumption n ≤ 2p + 2 implies τ ≤ p.Since we are trying to prove the existence of regular boundary points using valuesof β as small as possible, without loss of generality we can assume that β is veryclose to 1− 2/n = τ/(τ + 1), say

(7.6) β =τ(1 + ε)τ + 1

for some small ε > 0 to be eventually chosen. Now (5.5) amounts to prove that

(7.7)γτ

τ − β ≤ q =np

n− 2+ δi .

Substituting (7.6) in the previous inequality, since γ ≤ p, (7.7) is implied by

τ + 1τ − ε ≤

n

n− 2+δip.

Therefore, choosing ε ≡ ε(δi) ≡ ε(n,N, p, L/ν) small enough - recall the definitionof δi in (4.5) - we are reduced to prove that

(7.8)τ + 1τ≤ n

n− 2,

which in fact holds as an equality by (7.5). Therefore (6.3) holds with the choicein (7.5). We are now able to apply Proposition 6.1, with β as in (7.6), with thechosen ε. As a result we get that Du ∈ W 2θ

p ,p(Q+d ,RN ), for every θ < σv/2 and

every d < 1. Again Proposition 3.1 yields dimH(Σ0b) ≤ n − θ for every θ < σv/2,

and the definition of Hausdorff dimension implies dimH(Σ0b) ≤ n− σv. Recall that

σv ≡ σv(τ) has been defined in (5.7), with now τ being the one in (7.5). Thereforeit remains to check that

(7.9) q + p− 2γ > 1

and

(7.10) 2α > 1 andτβ

τ − β > 1 ,


to conclude with

(7.11) σv > 1

and hence with (7.4). The former inequality in (7.10) follows by the first assumptionin (1.33), while the latter follows from (7.6). As for (7.9), since γ ≤ p we noticethat

(7.12) q + p− 2γ ≥ q − p > np

n− 2− p =

2pn− 2

≥ 1 ,

where in the last inequality we have used n ≤ 2p+ 2. The proof of 7.2 is completefor the case n ≥ 6. In the case n ∈ 3, 4, 5, when assumption (1.33) gives β > 2/3,since 1 − 2/n < 2/3; therefore this time we take β = 2(1 + ε)/3 and we chooseτ = 2 in Proposition 6.1. Again (6.3) is implied by (7.8) which this time means tocheck that 3/(2− ε) ≤ 5/2 ≤ n/(n− 2), which of course holds for ε small. We thenproceed as for the case n ≥ 6 and the proof of Theorem 1.3 is complete as far as theassertion on the boundary regular points is concerned. As far the proof of (1.21) isconcerned, let us notice that using a standard compactness/covering argument wehave proved that

(7.13) Du ∈W 1p +ε,p(Ω ∩A,RNn) ε <

σv − 1p

,

where A is a neighborhood of ∂Ω. On the other hand we notice that Proposition6.1 holds in the interior; this means that the fractional differentiability result (6.4)holds for the original minimizer u of the Dirichlet problem (1.1) provided the cubeQ+

1 is obviously replaced by any interior cube Q b Ω. As a consequence, proceedingas above, we have

(7.14) Du ∈W1p +ε,p

loc (Ω,RNn) ε <σv − 1p

and this, together with (7.13) gives (1.21). The proof of Theorem 1.3 is finished.The proof of Theorem 1.4 proceeds in a similar manner; we first get Du ∈

Lq(Q+d ,RNn) for every d < 1 this time applying Proposition 8.3 since now we have

γ < p; then the rest of the proof goes as for Theorem 1.3, since from this pointon there we only used the weaker condition n ≤ 2p + 2, which is in fact the mainassumption of Theorem 1.4. ¤

Proof of Theorem 1.5. Let us first notice that in view of the application of Propo-sition 6.1, when looking at assumptions (1.33) we shall select τ ∈ [2, p] such that

(7.15) s =τ

τ + 1.

As in the previous proof we can assume that β is close to s, say β = s(1 + ε),that is again (7.6), for some small ε > 0 to be chosen; we now first observe thatγ ≤ ps + 2ps/(n − 2) implies (6.3) for τ determined in (7.15). Indeed by thedefinition in (4.6) we have to check that

γ ≤ np

n− 2

(τ − ετ + 1

)+ δi

(τ − ετ + 1

)

holds. Again by choosing ε ≡ ε(δi) small this is implied by

γ ≤ nps

n− 2= ps+

2psn− 2

,

which is our assumption on γ. Next, we observe that γ ≤ ps + 2ps/(n − 2) alsoimplies (7.9). Indeed, using the bound on γ, inequality (7.9) is implied by

(7.16)[

n

n− 2+ 1

]p ≥ 1 + 2ps+

4psn− 2

.


Now we distinguish two cases. In the first case we assume that n > 2p + 2. Thensince s ≤ p/(p+ 1), inequality (7.16) is implied is in turn implied by

2p ≥ 1 +2p2

p+ 1+

2pn− 2

[2pp+ 1

− 1].

Since we have that n ≥ 2p+ 2 then (7.16) is implied by

2p ≥ 1 +2p2

p+ 1+

[2pp+ 1

− 1]

which reveals to be an equality. In the second case we have n ≤ 2p + 2. Here wemay assume that

(7.17) s ≤ 1− 2n

otherwise Theorems 1.5 follows from Theorem 1.4. Using (7.17) we conclude, since(7.16) is implied by[

n

n− 2+ 1

]p ≥ 1 +

2p(n− 2)n

+4pn⇐⇒ 2p

n− 2≥ 1⇐⇒ n ≤ 2p+ 2 .

Therefore (7.9) follows in any case.Now we go back to Proposition 6.1, and we want to use it with the number τ ∈

[2, p] determined by (7.15), and β has been selected in (7.6); we have already seenthat (6.3) holds. It remains to check that Du ∈ Lq(Q+

d ,RNn) for every d < 1, butthis in turn follows by Proposition 8.3 below, since in Theorem 1.5 we are assumingthat γ < p. We can therefore apply Proposition 6.1, yielding Du ∈W 2θ

p ,p(Q+d ,RN ),

for every θ < σv/2 and every d < 1. Again Proposition 3.1 yields dimH(Σ0b) ≤ n−θ

for every θ < σv/2, and then dimH(Σ0b) ≤ n − σv. It remains to check that (7.11)

holds, where this time σv ≡ σv(τ) is determined by (5.7), and the number τ is fixedin (7.15). We have already seen that (7.9) holds, while as for (7.10), 2α > 1 isimplied by (1.33), and the second inequality is a consequence of the fact that weare taking β as in (7.6). Therefore σv > 1, and the proof is concluded. ¤

Proof of Theorem 1.8. We first remark that in the case of minima of functionals asin (1.43), under the assumption considered in Theorem 1.8, the partial regularityof minima and the regularity criterium (7.3) can be found in [6]. Then the proofproceeds largely as for Theorem 1.5. This time we use Proposition 8.5 to inferDu ∈ Lq(Q+

d ,RNn) whenever d < 1. This allows to apply Proposition 6.2, since(5.5) holds as for Theorem 1.5. We conclude that Du ∈W 2θ

p ,p(Q+d ,RN ), for every

θ < σ0/2 and every d < 1; here σ0 has been defined in (5.26). Again usingProposition 3.1 it follows that dimH(Σ0

b) ≤ n − σ0. Finally, the assumptions ofTheorem 1.8 imply that σ0 > 1, in turn yielding (7.4) and eventually (1.21) as forTheorem 1.3, and the proof is complete. ¤

Proof of Theorems 1.10-1.11. The proof parallels that for the case of functionals.As explained at the beginning of this section we can restrict to the case of partiallyDirichlet problems as (5.53), i.e.

div a0(x, u,Du) = 0 in Q+

1

u = 0 on Γ1 ,

and

(7.18) a0(x, y, z) = a(ψ(x), u0(ψ(x))+y,Du0(ψ(x))+zD(ψ−1)(x))(D(ψ−1)(x))T .

As usual, the new vector field a0(·) still satisfies the assumptions of Theorems1.10-1.11 for new constants ν, L depending on the chart used for flattening theboundary, and here u ≡ u ψ; moreover the characterization of the singular set


in (7.3) still holds; see [33, Section 3] and [17, Section 5] for details. We aretherefore reduced to establish (7.4). As for Theorem 1.10, since γ < p we applyProposition 8.4 below in order to have Du ∈ Lq(Q+

d ,RNn) whenever d < 1, andwhere q has been defined in (4.6). Observe that here we do not have to checkthe validity of (5.5) and therefore we can immediately use Proposition 6.3 in orderto obtain that Du ∈ W

2θp ,p(Q+

d ,RN ), for every θ < σs/2 and every d < 1; thenProposition 3.1 to get that dimH(Σ0

b) ≤ n − σv, where σv has been defined in(5.50). In turn assumptions (1.47) imply σs > 1, and the proof of Theorem 1.10 iscomplete. As for Theorem 1.11, we proceed like for Theorem 1.3, using Theorem4.1 instead of Theorem 4.2 to reduce to the case that u is Holder continuous up tothe boundary as for Theorem 1.3. Therefore applying Proposition 8.2 we have againthat Du ∈ Lq(Q+

d ,RNn) whenever d < 1 and therefore we can apply Proposition6.3 again. Finally, we need to check that σs > 1, and ultimately that (7.9) holds;for this we argue as for (7.12) and the proof is complete. ¤Remark 7.1 (Miscellanea of improvements). A careful examination of the aboveproofs reveals that the assumption n ≤ p + 2 can be relaxed to n ≤ p + 2 + δ foran absolute δ > 0 essentially depending n,N, p, L/ν. Moreover, we observe that(1.30)5 can be replaced by the following weaker condition:

|Fz(x, y1, z)− Fz(x, y2, z)| ≤ Lωα(|y1 − y2|)(1 + |z|2) γ−12 .

In fact, when estimating (I) in (5.14), we can use directly the last condition insteadof using first (1.30)5 and the estimate ωβ(·) ≤ ωα(·). Finally, by reformulating theresult in (1.21) in terms of the function V (·) defined in (2.5) we can improve (1.21)in

(7.19) V (Du) ∈W 1+ε2 ,2(Ω,RNn) and V (Du)|∂Ω ∈W ε,2(∂Ω,RNn)

hereby exploiting the usual effect linked to the use of the V -map: passing from(1.21) to (7.19) leads to a gain in differentiability and a loss in integrability - recallthat we are assuming p ≥ 2.

Remark 7.2. As already noticed in the proof of Theorem 1.3, the estimates foundfor the boundary singular set obviously hold in the interior too; moreover for suchinterior results we obviously do not need the boundary information u ≡ u0. Asa consequence (7.14) holds for any local minimizer u of the functional F [·] underthe assumptions of Theorems 1.3-1.5. Therefore, denoting by Ωu the set of interiorregular points of u according to (1.9), we have that

dimH(Ω \ Ωu) < n− 1 .

Such an estimate cannot be achieved by the methods of [36].

8. Variational Calderon-Zygmund estimates

Here we shall give the proof of the higher integrability results used in the previoussections to prove the global higher differentiability results. In all the various casesthat we have considered so far we will be able to satisfy assumption (6.1) of Section6. Some of the results presented here can be manipulated in order to cover acertain number of model problems which where not treated before and that inview of the recent literature devoted to the subject [10, 15, 21, 47], could havetheir own interest. These results improve those in [36, Section 7] and are obtainedby different arguments. On the other hand, in order to keep the treatment ata reasonable length, at several points we shall appeal to some of the argumentsof [3, 36, 26]. As usual, in the following, all the quantities representing radii orsidelengths, such as Rε, R0, will be smaller than one. We also remind the readerabout the convention for cubes adopted in Section 2.1. Finally, for the convenience


of the reader, we explicitly observe that, unless otherwise stated, all the constantsin this section will depend only on n,N, p, L/ν.

8.1. Low dimensions estimates. The results presented here were used in theproofs of the theorems, such as 1.2, 1.3 and 1.11, where the low dimension condition2 < n ≤ p + 2 is assumed. Therefore the assumptions considered will reflect theregularity properties held by solutions in such cases - compare with Section 4.1.We start with a problem of the type

(8.1)

∫

Q+1

G(x, u,Du) dx ≤∫

Q+1

G(x,w,Dw) dx ∀ w ∈ u+W 1,p0 (Q+

1 ,RN )

w ≡ u ≡ 0 on Γ1 ,

where

(8.2) G(x, y, z) = F0(x, h(x) + y,H(x) + z) .

while H(·) ∈ L∞(Q+1 ,RNn), where the exponent q has been defined in (4.6). Here

the integrand F0(·) satisfies (1.30)1,2 and (1.32). We notice that the integrandG(·) defined in (8.2) is of the type that arise after flattening the boundary inproblems of the type (1.1) - see (7.1) - where we identify h(x) ≡ u0(ψ(x)) andH(x) ≡ Du0(ψ(x)), while the presence of (Dψ)−1(x) has been incorporated in thex-dependence - recall that in Section 7 ψ is bi-C1,α.

Proposition 8.1. Let u ∈ W 1,p(Q+1 ,RN ) be a solution to the problem (8.1) and

assume that 2 < n ≤ p+ 2 together with

(8.3) Du ∈ Lp,µ1(Q+1 ,R

Nn) , u ∈ C0,λ1(Q+1 ,R

N ) λ1 > 0 , µ1 ∈ (2, n) ,

and

(8.4) h ∈ C0,α(Q1,RN ) and H ∈ L∞(Q+1 ,R

Nn) .

Then u ∈W 1,q(Q+d ,RN ), whenever d < 1 and Qd is concentric to Q1; the exponent

q has been defined in (4.6).

For the proof we start by observing that given ωσ(·) as in (1.32) and ε > 0, wecan determine Rε ≡ Rε(ε, σ, α, λ1, [h]C0,α , [u]C0,λ1 ) ∈ (0, 1) such that

(8.5) ωσ (Rε + oscRε(h) + oscRε(u)) ≤ ε .Here we are adopting the standard notation

oscRε(w) := supQs⊂Q+

1 ,s≤Rε

supx1,x2∈Qs

|w(x1)− w(x2)| w = h, u ,

where Qs denotes the general sub-cube of sidelength 2s contained in Q+1 . Next, we

recall a few higher integrability results of Gehring’s type for u. Since F0(·) in (8.2)in particular satisfies (1.30)1, we notice that the function G(·) in turn satisfies thefollowing growth conditions:

ν21−p|z|p − ν|H|p ≤ G(x, y, z) ≤ L2p(1 + |z|)p + L2p|H|p .By [25, Remark 6.6] we infer that the map u is a so called cubical quasi-minimizerof the functional

w 7→∫

Q+1

(1 + |Dw|p + |H|p) dx ,

for a suitable constant L ≡ L(n,N, p, L/ν) > 1. This means that∫

Q+1

(1 + |Du|p + |H|p) dx ≤ L∫

Q+1

(1 + |Dw|p + |H|p) dx


holds whenever w ∈ u+W 1,p0 (Q+

1 ,RN ). Therefore, by [25, Theorem 6.7], we inferthe existence of a further higher integrability exponent q3 > p, such that

(8.6)(−∫

Q

|Du|q3 dx

) 1q3 ≤ c

(−∫

λQ

(1 + |Du|p) dx) 1

p

+ c

(−∫

λQ

|H|q3 dx

) 1q3

,

holds whenever λ ∈ (1, 2], and whenever Q is a cube or a rectangle contained inQ+

1 ; the constant c depends only on n,N, p, L/ν, λ. Moreover if Q+ ⊂ Q+1 is an

upper cube such that (2Q)+ ⊂ Q+1 we have that the standard boundary version of

Gehring’s lemma implies

(8.7)(−∫

Q+|Du|q3 dx

) 1q3 ≤ c

(−∫

(λQ)+(1 + |Du|p) dx

) 1p

+c

(−∫

(λQ)+|H|q3 dx

) 1q3

.

For future convenience, let us put order in all such integrability exponents: bypossibly decreasing the values of q2, q3, we shall without loss of generality assumethat

(8.8) p < q2 < q3 < q = q1 − δi < q1 =np

n− 2+ 2δi .

See Section 4.2 for more definitions and notation.For the next decay type lemma, which is at the heart of all the subsequent results

in this section, we are using the convention about cubes established in Section 2.2.In particular all cubes considered will have sides parallel to the coordinate axes. Weconsider a cube QR0 ⊂ Q+

1 , clearly QR0 is not centered at the origin now. Finally,in the following the number χ will be such that χ ∈ 0, 1.Lemma 8.1. Let u ∈ W 1,p(Q+

1 ,RN ) be a solution to (8.1) under the assumption(8.3)-(8.4) and 2 < n ≤ p + 2. Let B > 1; there exists a positive number ε0 ≡ε0(n,N, p, L/ν,B), and a positive radius

(8.9) R0 ≡ R0(n,N, p, L/ν,B, σ, α, λ1, µ1, [h]C0,α , [u]C0,λ1 , ‖Du‖Lp,µ1 , ‖H‖L∞)

with the following property: If λ > 1 and Q ⊂ QR0/2 is a dyadic sub-cube of QR0/2

such that ∣∣∣Q ∩x ∈ QR0/2 : M∗((1 + |Du|+ χ|u|)p)(x) > ABλ ,

M∗((1 + |H|2) p2 )(x) < ε0λ

∣∣∣ > B−q1p |Q|(8.10)

holds, then its predecessor Q satisfies

(8.11) Q ⊆ x ∈ QR0/2 : M∗((1 + |Du|+ χ|u|)p)(x) > λ .Here M∗ ≡M∗

QR0denotes the restricted maximal function operator relative to QR0

in the sense of (2.6), and A ≡ A(n,N, p, L/ν) > 1 is an absolute constant; thenumber q1 > q has been defined in (4.5), while q has been defined in (4.6).

Proof. The proof, which is slightly more elaborate than strictly needed because weaim to adapt it to other cases later, goes by contradiction. The constants A,R0, ε0will be chosen towards the end of the proof. In a similar manner, in the followingε ∈ (0, 1) denotes a free parameter, and every time we fix a value of ε we denoteby Rε a radius such that (8.5) holds. The parameter ε will be diminished severaltimes, accordingly, Rε will be diminished too. We shall take

(8.12) R0 := Rε

and the value of R0 making the statement true will be determined towards the endof the proof, after, as mentioned, diminishing several times the values of ε. At thisstage Rε is initially determined according to (8.5), so that in the next lines all thedyadic sub-cubes of QR0/2 will have sidelength less than or equal than Rε/2.


Suppose now that (8.11) is not satisfied although (8.10) holds. We concludethere exists x ∈ Q such that

M∗((1 + |Du|+ χ|u|)p)(x) ≤ λholds; therefore, since Q ⊂ 3Q ⊂ QR0 because Q is the predecessor of Q, we have

(8.13) −∫

3Q

(1 + |Du|+ χ|u|)p dx ≤ λ .

By (8.10) we can find x1 ∈ Q such that M((1 + |H|)p)(x1) ≤ ε0λ and therefore

(8.14) −∫

3Q

(1 + |H|)p dx ≤ ε0λ .

From now on we set

(8.15) Qc :=52Q .

Now we define v ∈ u+W 1,p0 (Qc,RN ) as the unique minimizer of the functional

(8.16) w 7→ −∫

Qc

F0(x0, h(x0) + (u)Qc , Dw) dx ≡ −∫

Qc

K(Dw) dx ,

in the Dirichlet class u + W 1,p0 (Qc,RN ), with x0 denoting the center of Qc. The

minimality of v, the growth conditions (1.30)1 satisfied by F0(·), and (8.13), yield

(8.17) −∫

3Q

(1 + |Dv|)p dx ≤ c(p, L/ν)−∫

3Q

(1 + |Du|)p dx ≤ cλ .

Next, we recall that v solves the Euler-Lagrange system divKz(Dv) = 0 in Qc,which by (1.30)1,2 is of the type considered in Section 4.2 and (4.3). Therefore, by(4.7) used with B ≡ Qc and λ = 5/4, by (8.13), (8.17), and by Poincare’s inequalityand recalling that u ≡ v on ∂(3Q), we get

(−∫

2Q

(1 + |Dv|+ χ|v|)q1 dx

) 1q1 ≤ c

(−∫

Qc

(1 + |Dv|+ χ|v|)p dx

) 1p

≤ c

(−∫

3Q

(1 + |Du|+ χ|u|)p dx

) 1p

≤ cλ 1p ,(8.18)

with c ≡ c(n,N, p, L/ν). Moreover, using (4.10) with Q ≡ Qc, we have, withq2 ∈ (p, q3)

(−∫

Qc

(1 + |Dv|)q2 dx

) 1q2 ≤ c

(−∫

Qc

(1 + |Dv|)p dx

) 1p

+ c

(−∫

Qc

|Du|q2 dx

) 1q2

≤ cλ1p +

(−∫

Qc

|Du|q2 dx

) 1q2

,(8.19)

where we used also (8.17). Moreover using (8.6) with Q ≡ Qc, λ ≡ 6/5, and (8.8)to employ Holder’s inequality, we have

(−∫

Qc

|Du|q2 dx

) 1q2 ≤ c

(−∫

3Q

(1 + |Du|p) dx) 1

p

+ c

(−∫

3Q

(1 + |H|)q3 dx

) 1q3

≤ cλ1p + c(1 + ‖H‖L∞)

q3−pp

(−∫

3Q

(1 + |H|)p dx

) 1p

≤ c(‖H‖L∞)λ1p .(8.20)


We also used (8.13)-(8.14). Combining this last estimate with (8.19) we gain

(8.21) −∫

Qc

(1 + |Dv|)q2 dx ≤ c(‖H‖L∞)λq2p .

Again using divKz(Dv) = 0, and Taylor’s formula similarly to (5.10), we obtain

1c−∫

Qc

|Du−Dv|p dx

≤ −∫

Qc

(K(Du)−K(Dv)− 〈Kz(Dv), Du−Dv〉

)dx

= −∫

Qc

(K(Du)−K(Dv)

)dx

= −∫

Qc

(K(Du)−G(x, u,Du)

)dx+ −

∫

Qc

(G(x, u,Du)−G(x, v,Dv)

)dx

+ −∫

Qc

(G(x, v,Dv)−K(Dv)

)dx =: (I) + (II) + (III)

≤ (I) + (III) .(8.22)

In the last line we used the minimality of u to deduce (II) ≤ 0. Using (1.32) and(8.12) we have

(I) ≤ c−∫

Qc

ωσ (Rε + oscRε(h) + oscRε(u)) (1 + |Du|)p dx

+c−∫

Qc

(1 + |H|+ |Du|)p−1|H| dx =: (IV ) + (V ) .

Note that in the last estimate we have used the convexity of z 7→ F0(·, z) whichimplies the growth condition |(F0)z(z)| ≤ c(L/ν)(1 + |z|p−1) - see [13, Proposition2.32]. In turn (8.5) and (8.13) give

(IV ) ≤ cε −∫

Qc

(1 + |Du|)p dx ≤ cελ ,

while by Young’s inequality and (8.13) we have

(V ) ≤ ε−∫

Qc

(1 + |Du|)p dx+ c(ε)−∫

Qc

(1 + |H|)p dx ≤ cελ+ c1(ε)ε0λ .

The estimation of (III) is slightly more delicate. Using (1.32) satisfied by F0(·),again (8.12), and adding and subtracting u(x) inside ωσ(·), and finally using thedefinition of ωσ(·) in (1.32) with (2.4), yields

(III) ≤ c−∫

Qc

ωσ (Rε + oscRε(h) + oscRε(u)) (1 + |Dv|)p dx

+c−∫

Qc

ωσ (|v − u|) (1 + |Dv|)p dx

+c−∫

Qc

(1 + |H|+ |Dv|)p−1|H| dx =: (V I) + (V II) + (V III) .

As for (IV ), (V ), and using also (8.5), (8.17) and Young’s inequality, we have

(V I) + (V III) ≤ c[ελ+ c1(ε)ε0]λ .

As for (V II), using Holder’s inequality, the definition of ωσ(·) ≤ 1 in (1.32) and inparticular the concavity of ωσ(·) to use Jensen’s inequality in a standard way, and


taking (8.17) and (8.21) into account, we have

(V II) ≤ c

(−∫

Qc

ωσ(|v − u|)q2

q2−p dx

)1− pq2

(−∫

Qc

(1 + |Dv|)q2 dx

) pq2

≤ cωσ

(−∫

Qc

|v − u| dx)1− p

q2

λ

≤ cωσ

[(−∫

Qc

|v − u|p dx) 1

p

]1− pq2

λ

≤ cωσ

[(|Qc|

pn −

∫

Qc

|Dv|p + |Du|p dx) 1

p

]1− pq2

λ

≤ cωσ

[(|Qc|

pn −

∫

Qc

(1 + |Du|p) dx) 1

p

]1− pq2

λ

≤ cωσ

[(|Qc|

p+µ1−nn (1 + ‖Du‖)Lp,µ1

) 1p

]1− pq2

λ

≤ cωσ

(|Qc|

µ1−2np

)1− pq2λ

≤ cξ(Rε)λ ≡ ch(‖Du‖Lp,µ1 , ‖H‖L∞)ξ(Rε)λ ,

with

ξ(s) := ωσ

(s

µ1−2p

) q2−pq2

.

Note that in the last estimate we used again that n ≤ p+ 2; moreover it is crucialthat µ1 > 2 and q2 > p hold, in order to have that ξ(s)→ 0 when s→ 0. Mergingthe estimates found for (I), . . . , (V III) with (8.22) we finally obtain

(8.23) −∫

Qc

|Du−Dv|p dx ≤ c [ε+ ch(‖Du‖Lp,µ1 , ‖H‖L∞)ξ(Rε) + c1(ε)ε0]λ ,

and recall that c1(ε) → ∞ when ε → 0, while c ≡ c(n,N, p, L/ν); note that inthe whole proof up to now the only constant depending on one of the quantities‖Du‖Lp,µ1 , ‖H‖L∞ is actually ch.

We are now ready for the final comparison argument. We denote by M∗∗ themaximal operator restricted to the cube 2Q - see (2.6). We have

|x ∈ Q : M∗∗((1 + |Du|+ χ|u|)p)(x) > ABλ|≤ |x ∈ Q : M∗∗((1 + |Dv|+ χ|v|)p)(x) > 8−pABλ|

+|x ∈ Q : M∗∗((|Du−Dv|+ χ|u− v|)p)(x) > 8−pABλ|≤ c(n, q1, p)

(ABλ)q1p

∫

2Q

(1 + |Dv|+ χ|v|)q1 dx

+c(n, q1, p)ABλ

∫

2Q

(|Du−Dv|p + χ|u− v|p) dx ,(8.24)

where in the last line we made use of (2.7) with t ≡ q1/p > 1. Using (8.18) we canestimate

c(n, q1, p)

(ABλ)q1p

∫

2Q

(1 + |Dv|+ χ|v|)q1 dx ≤ c

(AB)q1p

|Q| ,

where c ≡ c(n,N, p, L/ν) > 1. By choosing

(8.25) A := 8n+2(c+ 100n) =⇒ A ≡ A(n,N, p, L/ν)


we obtain

(8.26)c(n, q1, p)

(ABλ)q1p

∫

2Q

(1 + |Dv|+ χ|v|)q1 dx ≤ 1

8n+2Bq1p

|Q| ,

and this fixes the constant A, yielding the absolute dependence upon the constantsmentioned in the statement of the Lemma. We now turn to the last integral in(8.24). Poincare’s inequality and (8.23) give

c(n, q1, p)ABλ

∫

2Q

(|Du−Dv|p + χ|u− v|p) dx

≤ c [ε+ ch(‖Du‖Lp,µ1 , ‖H‖L∞)ξ(Rε) + c1(ε)ε0]|Q|AB

(8.27)

with c ≡ c(n,N, p, L/ν). We take ε ≡ ε(n,N, p, L/ν,B) > 0 small enough to get

(8.28)cε

A≤ 1

8n+2Bq1p −1

.

Observe that this determines a first tail of smallness for

Rε ≡ Rε(n,N, p, L/ν,B, σ, α, λ1, [h]C0,α , [u]C0,λ1 )

- we recall that Rε has been implicitly defined through (8.5). Next we can takeε0 ≡ ε0(n,N, p, L/ν,B) > 0 according to

(8.29)cc1(ε)ε0

A≤ 1

8n+2Bq1p −1

.

Next we further diminish the value of Rε according to

(8.30)cch(‖Du‖Lp,µ1 , ‖H‖L∞)ξ(Rε)

A≤ 1

8n+2Bq1p −1

.

This fixes Rε ≡ Rε(n,N, p, L/ν,B, σ, α, λ1, µ1, [h]C0,α , [u]C0,λ1 , ‖Du‖Lp,µ1 , ‖H‖L∞)and this finally determines the value of R0 via (8.12), with the prescribed depen-dence upon the various constants. Connecting (8.28)-(8.30) to (8.27) we gain

(8.31)c(n, q1, p)ABλ

∫

2Q

(|Du−Dv|p + χ|u− v|p) dx ≤ |Q|8n+2B

q1p

.

In turn using this last inequality and (8.26) together with (8.24) we have

(8.32) |x ∈ Q : M∗∗((1 + |Du|+ χ|u|)p)(x) > ABλ| ≤ 8−n−1B−q1/p|Q| .To conclude we observe that (8.13) implies

M∗((1 + |Du|+ χ|u|)p)(x) ≤ maxM∗∗((1 + |Du|+ χ|u|)p)(x), 100nλ ,for every x ∈ Q and therefore, since AB > 100n by (8.25) and (8.32) we gain

|x ∈ Q : M∗((1 + |Du|+ χ|u|)p)(x) > ABλ| ≤ (1/2)B−q1/p|Q| ,which is a contradiction to (8.10). The proof of the Lemma is complete. ¤

The previous lemma has a boundary companion. In the following QR0 ⊂ Q1

denotes a cube centered on xn = 0; recall the notation introduced in Sections 2.1and 2.2.

Lemma 8.2. Let u ∈ W 1,p(Q+1 ,RN ) be a solution to (8.1) under the assumption

(8.3)-(8.4) and 2 < n ≤ p + 2. Let B > 1; there exists a positive number ε0 ≡ε0(n,N, p, L/ν,B), and a positive radius as in (8.9), with the following property: Ifλ > 1 and R ⊂ (QR0/2)+ is a dyadic sub-rectangle of (QR0/2)+ such that

∣∣∣R∩x ∈ (QR0/2)+ : M∗((1 + |Du|+ χ|u|)p)(x) > ABλ ,

M∗((1 + |H|2) p2 )(x) < ε0λ

∣∣∣ > B−q1p |R|


holds, then its predecessor Q satisfies

Q ⊆ x ∈ (QR0/2)+ : M∗((1 + |Du|+ χ|u|)p)(x) > λ .Here M∗ ≡ M∗

(QR0 )+ denotes the restricted maximal function operator relative to(QR0)

+ in the sense of (2.9), and A ≡ A(n,N, p, L/ν) > 1 is an absolute constant;the number q1 > q has been defined in (4.5), while q has been defined in (4.6).

The proof of the previous result can be obtained by matching the proof of Lemma8.1 with the one of [36, Lemma 7.5]. Observe that all the a priori estimates for thesolutions of the comparison problems used for Lemma 8.1 work on full upper cubesand rectangles, taking into account the boundary datum too; (4.17) must be usedin (8.18), (8.7) must be used in (8.20). We refer to [36] for more details.

Proof of Proposition 8.1. With Lemmas 8.1-8.2 having been established the fullproof of Proposition 8.2 can be obtained following [36, Section 7], see also [3, 26].We summarize the main steps for later convenience. One chooses B large enoughin order to have

(8.33)1

Bq1p −1

=1

2Aqp

- here it is important to have q1 > q, compare with (4.6) - and therefore it followsthat B ≡ B(n,N, p, L/ν) since also δi defined in (4.5) depends on the same pa-rameters and q1 − q = δi, while A is in (8.25). Now, thanks to Lemmas 8.1-8.2 itis possible to apply iteratively Proposition 2.1 on level sets of the type (AB)kλ0,k ∈ N, where

λ0 := LB q1p −

∫

Q

(1 + |Du|+ χ|u|)p dx , Q = QRε , Q+Rε

,

with L suitably large. Compare also with [3, Theorem 1] and [26, Section 8.5]. Suchan iteration method finds its origins in [10]. The final outcome of this procedure isthe following reverse Holder type inequality:

(−∫

Q

(|Du|+ χ|u|)q dx

) 1q

≤ c

(−∫

λQ

(1 + |Du|+ χ|u|)p dx

) 1p

+c(−∫

λQ

(1 + |H|)q dx

) 1q

,(8.34)

which holds whenever λ ∈ (1, 2] and Q is either a cube such with sidelength lessthan or equal than 2R0, and c depends only on the quantities n,N, p, L/ν, λ, whileχ ∈ 0, 1. Iterating Lemma 8.2 as in [36, Section 7] we instead have that

(−∫

Q

(|Du|+ χ|u|)q dx

) 1q

≤ c

(−∫

(λQ)+(1 + |Du|+ χ|u|)p dx

) 1p

+c

(−∫

(λQ)+(1 + |H|)q dx

) 1q

,(8.35)

holds whenever Q is a cube centered on xn = 0 and such that 2Q ⊂ Q1, with thesame meaning for c, λ. Notice that at this stage R0 has been determined essentiallydepending on the parameters stated in Lemma 8.1 and via (8.33). The proof ofProposition 8.1 now follows via a standard covering argument. We also notice thatfollowing the arguments of [3, 36] one arrives at (8.34)-(8.35) with λ = 2; the caseof λ ∈ (1, 2] then follows by a standard covering/iteration argument - compare with[25, Chapter 6]. ¤


Remark 8.1. Assuming n ≤ p + 2 and Du ∈ Lp,µ1 for some µ1 > 2 actuallyimplies u ∈ C0,λ1 for some λ1 > 0; this is an easy consequence of Campanato-Meyers integral characterization of Holder continuity. Therefore in Proposition 8.1we may also avoid assuming that u ∈ C0,λ1 . On the other hand we preferred toinclude such an assumption to facilitate the reading of the forthcoming proofs.

We now give the analog of Proposition 8.1 for systems, whose proof turns out tobe considerably simpler. We consider a problem of the type

(8.36)

div b(x, h(x) + u,H(x) +Du) = 0 in Q+1

u = 0 on Γ1 ,

for some R > 0, where the vector field b(·) satisfies (1.44)1,2,4 and (1.45). Againsuch kind of problems naturally arise form the standard flattening of the boundaryprocedure to treat systems as (1.6) - compare with (7.18).


assume that u ∈ C0,λ1(Q+1 ,RN ) for some λ1 > 0, together with

(8.37) h ∈ C0,α(Q1,RN ) and H(·) ∈ Lq(Q+1 ,R

Nn) .

Then u ∈W 1,q(Q+d ,RN ), whenever d < 1 and Qd is concentric to Q1.

Proof. The proof closely follows that of Proposition 8.1. In particular once theexact analogs of Lemmas 8.1-8.2 are achieved the rest of the proof follows verbatimthat of Proposition 8.1. We confine ourselves to give the modifications for Lemma8.1, those for Lemma 8.2 being entirely similar. We take χ = 0 here. One arrives at(8.15) and defines the comparison map v ∈ u+W 1,p

0 (Qc,RN ) as the unique solutionto the Dirichlet problem

(8.38)

divB(Dv) := div b(x0, h(x0) + (u)Qc , Dv) = 0 in Qc

v = u on ∂Qc .

First notice that (8.17) follows testing the weak formulation of (8.38)1 with u− v,and using the fact that (1.44)1,2 imply that

c(L/ν)−1|z|p − c(L/ν) ≤ 〈b(x0, h(x0) + (u)Qc , z), z〉holds whenever z ∈ RNn. Then, once a proper analog of (8.23) is established,the rest of the proof follows. We emphasize the fact that we are assuming neitherH ∈ L∞ nor Du ∈ Lp,µ1 . In turn, using the fact that v solves (8.38) and u solves(8.36), using also Young’s inequality we have

−∫

Qc

|Du−Dv|p dx ≤ c∫

Qc

〈B(Du)−B(Dv), Du−Dv〉 dx

≤ c−∫

Qc

〈B(Du)− b(x, h(x) + u,Du), Du−Dv〉 dx

+c−∫

Qc

〈b(x, h(x) + u,Du)− b(x, h(x) + u,H +Du), Du−Dv〉 dx

≤ c−∫

Qc

|b(x, h(x) + u,Du)− b(x, h(x) + u,H +Du)| pp−1 dx

+c−∫

Qc

|B(Du)− b(x, h(x) + u,Du)| pp−1 dx+

12−∫

Qc

|Du−Dv|p dx .(8.39)

By (8.5) and (1.45) satisfied by b(·), we have

(8.40) −∫

Qc

|B(Du)− b(x, h(x) + u,Du)| pp−1 dx ≤ cε p

p−1 −∫

Qc

(1 + |Du|)p dx ≤ cελ .


Using (1.44)1, satisfied by b(·), we have, using Young’s inequality with conjugateexponents (p− 1, (p− 1)/(p− 2)) when p > 2

−∫

Qc

|b(x, h(x) + u,H +Du)− b(x, h(x) + u,Du)| pp−1 dx

≤ c−∫

Qc

(1 + |H|+ |Du|) p(p−2)p−1 |H| p

p−1 dx

≤ cε−∫

3Q

(1 + |Du|)p−1 dx+ c1(ε)−∫

3Q

(1 + |H|)p dx ≤ cελ+ c1(ε)ε0λ .

Combining the last three estimates yields

(8.41) −∫

Qc

|Du−Dv|p dx ≤ cελ+ c1(ε)ε0λ ,

and the rest of the proof follows as after (8.23), via the comparison argument in(8.24).

but we just rely on the Holder continuity of the solution.¤

Remark 8.2. In the case of systems (8.38) the radius R0 > 0 actually exhibits thesimpler dependence

(8.42) R0 ≡ R0(n,N, p, L/ν, σ, α, λ1, [h]C0,α , [u]C0,λ1 ) .

Indeed, let us go back to the proof of Lemma 8.1: the only point where the norms‖Du‖Lp,µ1 , ‖H‖L∞ do appear is inequality (8.23), via the constant ch. This de-pendence eventually reflects in the choice of R0 in (8.30)-(8.12). In the case ofProposition 8.2 inequality (8.23) is replaced by (8.41), where ‖Du‖Lp,µ1 , ‖H‖L∞do not appear - neither they are assumed to be finite in Proposition 8.2. At thisstage we come up with R0 still depending on n,N, p, L/ν, σ, [h]C0,α , [u]C0,λ1 andB. Then one fixes B ≡ B(n,N, p, L/ν) according to (8.33), and finally plugs thisdependence into the one already found for R0 and then concludes with (8.42). Weconclude with another observation we shall eventually use. The choice in (8.5) isclearly motivated by the necessity of estimating terms involving the oscillations ofh and u, as the first integral appearing in (8.40). When in the special situation ofno dependence on the variable y of the energy F0(·) or of the vector field b(·) - thatis we have no dependence on u(x) as a coefficient is allowed - the choice in (8.5)can be simplified in

(8.43) ωσ (Rε) ≤ ε .As a consequence, in the choice of R0 operated in (8.12), u and h are not involved,and (8.42) turns into

(8.44) R0 ≡ R0(n,N, p, L/ν, σ) .

We finally observe that using estimates (8.34)-(8.35) and a standard covering argu-ment we have that

(−∫

Q+d

|Du|q dx) 1

q

≤ c(−∫

Q+1

(1 + |Du|)p dx

) 1p

+ c

(−∫

Q+1

(1 + |H|)q dx

) 1q

,

holds, whenever 0 < d < 1. The constant here depends only on n,N, p, L/ν and d,as an effect of the dependence on the constants identified in (8.44).


8.2. High dimensions estimates. Here we prove the higher integrability resultsnecessary to deal with the higher dimensional case of Theorems 1.5, 1.1 and ingeneral those valid without the restriction n ≤ p + 2. The point here is that therole played by (8.3), which holds when n ≤ p + 2, is played by the assumptionγ < p. The results are the following:


assume that F0(·) satisfies (1.30)1,2,3 with γ < p, and (1.32); moreover assume(8.37), where q is in (4.6). Then u ∈ W 1,q(Q+

d ,RN ), whenever d < 1 and Qd isconcentric to Q1.

The corresponding statement for systems is


assume that b(·) satisfies (1.44)1,2,4 with γ < p, and (1.45); moreover assume(8.37), where q is in (4.6). Then u ∈ W 1,q(Q+

d ,RN ), whenever d < 1 and Qd isconcentric to Q1.

Proof of Propositions 8.3-8.4. The proof of the two previous results parallels thoseof the analogous Propositions 8.1-8.2, and we shall give the necessary modifications,once again only for Lemma 8.1; we shall always take χ = 0. We start from the caseof functionals. Since here we are not assuming that the solution u is continuous,the definition of Rε, implicitly given via (8.5), must be modified in

(8.45) ωσ (Rε + oscRε(h)) +Rσmε ≤ ε ,

where

(8.46) σm := minσ, p− γ .This last number is positive since we are assuming γ < p. Then we proceed asin the proof of Lemma 8.1, arriving up to (8.18) and jumping directly to (8.22);we want to find an estimate of the type (8.41). Keeping the notation adopted forLemma 8.1, and using (1.30)2,3, (1.32) and (8.45) we have

(I) + (III) ≤ c −∫

Qc

ωσ (Rε + oscRε(h)) (1 + |Du|+ |Dv|)p dx

+c −∫

Qc

ωσ (|v − u|+ |u− (u)Qc |) (1 + |Du|+ |Dv|)γ dx

+c −∫

Qc

(1 + |H|+ |Du|+ |Dv|)p−1|H| dx =: (IX) + (X) + (XI) .(8.47)

Observe that we assumed without loss of generality that σ ≤ α, therefore ωβ(·) ≤ωσ(·). The estimation of (IX) and (XI) can be achieved as in Lemma 8.1: using(8.45), (8.17) and Young’s inequality to get

(IX) + (XI) ≤ cε −∫

Qc

(1 + |Du|+ |Dv|)p dx+ c1(ε) −∫

Qc

(1 + |H|)p dx

≤ c[ε+ c1(ε)ε0]λ .

It remains to estimate (X). We have, via Young’s inequality

(X) ≤ c|Qc|σmn −

∫

Qc

(|Qc|

−γ−σmn [ωσ (|v − u|+ |u− (u)Qc |)]

γ+σmσm

+(1 + |Du|+ |Dv|)γ+σm

)dx

≤ cRσmε −

∫

Qc

(|Qc|

−γ−σmn

(|v − u|γ+σm + |u− (u)Qc |γ+σm)


+(1 + |Du|+ |Dv|)γ+σm

)dx

≤ cε −∫

Qc

(1 + |Dv|+ |Du|)γ+σm dx ≤ cε −∫

Qc

(1 + |Du|)p dx ≤ cελ(8.48)

where we used Poincare’s inequality twice and (8.17),(8.45) - of course we also usedthat (8.46) implies γ + σm ≤ p. With such estimates replacing the analogous onesobtained in Lemma 8.1, we arrive at (8.41) once again, and the proof of Lemma 8.1in the present situation follows.

We now describe the modifications required to prove Proposition 8.4. Again asin the proof of Proposition 8.2 we arrive at (8.39), where the only term which mustbe estimated in a different way is the second-last one. Thereby, using (1.44)4 and(1.45) satisfied by b(·), we have

−∫

Qc

|B(Du)− b(x, h(x) + u,Du)| pp−1 dx

≤ c −∫

Qc

[ωσ (Rε + oscRε(h))]

pp−1 (1 + |Du|)p dx

+c −∫

Qc

[ωσ (|u− (u)Qc |)]p

p−1 (1 + |Du|) p(γ−1)p−1 dx =: (XII) + (XIII) .

In turn, using (8.45) and (8.13) we gain (XII) ≤ cελ, while as for (XIII), sinceγ ≤ p implies p(γ − 1)/(p − 1) ≤ γ, and since ωσ(·) ≤ 1, arguing as for (X) in(8.47)-(8.48) we have

(XIII) ≤ c −∫

Qc

ωσ (|u− (u)Qc |) (1 + |Du|)γ dx ≤ cελ .

Using this last estimate and (XII) ≤ cελ in (8.39) we again arrive at (8.41) andthe proof goes on as for Proposition 8.2. ¤

We finally give the corresponding results for treating functionals as in Theorem1.8. For this we consider a solution u ∈W 1,p(Q+

1 ,RN ) of the following minimizationproblem(8.49)

∫

Q+1

f(x,H +Du) + f1(x, h+ u) dx ≤∫

Q+1

f(x,H +Dw) + f1(x, h+ w) dx

w ≡ u ≡ 0 on Γ1 ,

for every w ∈ u+W 1,p0 (Q+

1 ,RN ). The assumptions are related to those of Theorem1.8; specifically, the integrand f1(·) is a Caratheodory function such that

(8.50)

0 ≤ f1(x, y) ≤ L(1 + |y|)γ

|f1(x, y1)− f1(x, y2)| ≤ Lωσ(|y1 − y2|)(1 + |y1|+ |y2|)γγ < p

hold whenever x ∈ Q+1 and y, y1, y2 ∈ RN , L ≥ 1.

Proposition 8.5. Let u ∈ W 1,p(Q+1 ,RN ) be a solution to the problem (8.49),

where f(·) satisfies (1.41)1,2 and (1.32), and where f1(·) satisfies (8.50). Assumeh ∈ L∞(Q+

1 ,RN ) and H ∈ Lq(Q+1 ,RNn). Then u ∈ W 1,q(Q+

d ,RN ), wheneverd < 1 and Qd is concentric to Q1, and where q has been defined in (4.6).

Proof. Again, it is sufficient to reprove Lemma 8.1 for the present situation, with asimilar statement that holds up to the boundary; we take χ = 1 here. As for (8.45)we define Rε to be a radius such that

(8.51) ωσ(Rε) + (1 + ‖h‖L∞)γRσmε ≤ ε .


Arriving at (8.16) we define v ∈ u+W 1,p0 (Qc,RN ) as the unique minimizer of the

functional

w 7→∫

Qc

f(x0, Dw) dx ,

in the Dirichlet class u+W 1,p0 (Qc,RN ), with x0 denoting the center of Q. We gain

(8.18), and then, using also the minimality of u we get

1c−∫

Qc

|Du−Dv|p dx ≤ −∫

Qc

(f(x0, Du)− f(x0, Du)

)dx

≤ −∫

Qc

(f(x0, Du)− f(x0, Du)

)dx+ −

∫

Qc

(f(x,H +Du) + f1(x, h+ u)

)dx

− −∫

Qc

(f(x,H +Dv) + f1(x, h+ v)

)dx

≤ c −∫

Qc

ωσ(|x− x0|)(1 + |Du|p + |Dv|p) dx

+c −∫

Qc

(1 + |Du|+ |Dv|+ |H|)p−1|H| dx

+ −∫

Qc

(f1(x, h+ u)− f1(x, h+ v)

)dx

≤ cε −∫

Qc

(1 + |Du|p) dx+ c1(ε) −∫

Qc

(1 + |H|)p dx

+c(1 + ‖h‖L∞)γ −∫

Qc

ωσ(|u− v|)(1 + |u|+ |v|)γ dx .(8.52)

In turn, by Young’s and Poincare’s inequalities, as for (8.48), but this time using(8.51), we have

(1 + ‖h‖L∞)γ −∫

Qc

ωσ(|u− v|)(1 + |u|+ |v|)γ dx

≤ c(1 + ‖h‖L∞)γRσmε

(−∫

Qc

|Qc|−γ−σm

n |v − u|γ+σm

+(1 + |u|+ |u− v|)γ+σm

)dx

≤ cε −∫

Qc

(1 + |Du|p + |Dv|p + |u|p) dx ≤ ε −∫

3Q

(1 + |Du|p + |u|p) dx ≤ cελ ,

where σm has been defined in (8.46); we have also used (8.17) and (8.13); here it isχ = 1. Combining the last estimate with (8.52), and recalling (8.14) we gain

−∫

Qc

(|Du−Dv|p + |u− v|p) dx ≤ c [ε+ c1(ε)ε0]λ ,

with c ≡ c(n,N, p, L/ν); we have used Poincare’s inequality. After such an inequal-ity we may proceed exactly as after (8.23) and the proof of the whole Proposition8.5 follows. ¤

8.3. Uniform estimates on small domains. The higher integrability resultsfrom the previous sections have been used essentially in a qualitative way - comparewith the proofs given in Section 7. Here we shall need precise a priori estimates onthe subdomains AR introduced in (5.34), and such estimates must be independentof R, since they have to be applied in the context of lemmas as 5.3 and 6.1. Onceproved the main result of this section, i.e. Proposition 8.6 below, Theorem 5.1 will


be a special case. Specifically, we consider Dirichlet problems as

(8.53)

div B(x,Du) = 0 in AR

u = u0 on ∂AR ,

with u0 ∈ W 1,q(AR,RN ), and q having been defined in (4.6). The domain AR isthe one introduced in Section 5.2, (5.33). On the vector field B : AR×RNn → RNn

we prescribe a continuous coefficients dependence i.e. for some σ > 0 we assume

(8.54) |B(x1, z)−B(x2, z)| ≤ Lωσ(|x1 − x2|)(1 + |z|)p−1 .

Proposition 8.6. Let u ∈ W 1,p(AR,RN ) be a solution to the problem (8.53) andassume that B(·) satisfies (4.12)1 and (8.54). Then there exists a constant, de-pending only on n,N, p, L/ν, ‖∂A1‖C1,1 , σ, but otherwise independent of R > 0, ofu0, u, and of the vector field B(·), such that

(8.55)∫

AR

|Du|q dx ≤ c∫

AR

(1 + |Du0|q) dx

holds, where q has been defined in (4.6). Moreover (8.55) holds for any ball BR ≡AR.

Proof. As usual, when considering the domain AR, we think that the ball BR

appearing in (5.34) is centered at the origin. We first observe that it suffices toreduce to the case of AR ≡ A1, by the way AR has been defined in (5.33). Indeedlet us introduce the scaled maps

(8.56) uR : x ∈ A1 7→ u(Rx)R

∈ RN uR0 : x ∈ A1 7→ u0(Rx)

R∈ RN .

It is then easy to see that ur solves the Dirichlet problemdiv B(Rx,DuR) = 0 in A1

uR = uR0 on ∂A1 .

Then we notice that (x, z) 7→ B(Rx, z) satisfies (4.12)1 and (8.54), with ωσ(·)replaced by ωσ(R·); since ωσ(·) is an increasing function and R ≤ 1 then we concludethat (x, z) 7→ B(Rx, z) still satisfies (8.54). Therefore assuming the validity of thetheorem for A1 we have that∫

A1

|DuR|q dx ≤ c∫

A1

(1 + |DuR0 |q) dx

holds, and (8.55) in the general case follows scaling back the previous inequality.Therefore from now one we have reduced to the case AR ≡ A1. In turn, afterreducing to the case of the zero boundary value by considering the new vector field(x, z) 7→ B(x,Du0(x) + z), we flatten the boundary ∂A1 via the charts describedin Remark (5.1). Call (ρi, Ci) ≡ (ρ,C) one of such charts, and set ψ ≡ ρ−1. Thenwe obtain a problem of the type

(8.57)

div b(x,H(x) +Du) = 0 in Q+1

u = 0 on Γ1 ,

where H(x) = Du0(ψ(x)); the vector field b(·) has been obtained by pulling backthe original vector field b(·,H + ·) via ψ i.e.

b(x,H(x) + z) = B(ψ(x), Du0(ψ(x)) + z(Dψ)−1(x))[(Dψ)−1(x)]T .

Compare with Section 7 and (7.18). Using the bounds in (5.37) together with thefact that B(·) satisfies (4.12)1 and (8.54), it is now easy to see that the vector fieldz 7→ b(·, z) (not z 7→ b(·,H + z)!!) satisfies (4.12)1 with new constants ν(M), L(M)- recall that the number M appears in (5.37) - and satisfies also

|b(x1, z)− b(x2, z)| ≤ c(L,M)ωσ(|x1 − x2|)(1 + |z|)p−1 ,


whenever x1, x2 ∈ Q+1 , and s ≥ 0; note indeed that we use that s ≤ cωσ(s) for s ≤ 1.

We are now in position to repeat the proof of Proposition 8.2 keeping Remark 8.2in mind, and since we are in the situation of a system with no dependence onu(x) in the coefficients, we come up with a final radius R0, which only dependson n,N, p, L/ν, σ,M as in (8.44). We therefore arrive at inequalities (8.34)-(8.35),which hold for cubes and upper cubes respectively, with sidelength not exceeding2R0, where the radius R0 depends only on n,N, p, L/ν, σ,M . Note here that thedependence of R0 on the geometry of ∂A1 comes from M via (5.37). The inequality(8.2), valid for solutions to flattened problems as (8.57), can be now transferred backto the original solution u of (8.53) via the charts (ρi, Ci) for i ≤ K. Specifically,using the area formula we find a finite number, K1 ≈ (R2R0)−nK ≈ K (thenumber R2 has been defined in the construction after (5.34) and depends only onthe geometry of A1) and a family of cubes Qii≤K1 , such that they are centeredon ∂A1, their sidelength is uniformly comparable to R0, their union covers ∂A1,and such that

(−∫

Q∩A1

|Du|q dx) 1

q

≤ c

(−∫

2Q∩A1

(1 + |Du|)p dx

) 1p

+c(−∫

2Q∩A1

(1 + |Du0|)q dx

) 1q

(8.58)

holds for every Q ≡ Qi and i ≤ K1. On the other hand notice that the proof ofLemma 8.1 - as modified above for the present situation - applies directly to thesystem (8.53)1 as far as interior (to A1) cubes are concerned and therefore (8.34)holds for cubes contained in A1 and with suitably small sidelength. All in all we findanother positive radius R3 ≡ R3(n,N, p, L/ν, σ) such that (8.58) holds wheneverQ is a cube of sidelength not exceed 2R3 such that 2Q ⊂ A1; there is of courseno loss of generality in assuming that R0 ≤ R3. We have covered A1 with a newfinite family of cubes, still denoted by Qii≤K2 , where again K2 ≈ cK for which(8.58) holds and c ≡ c(n,N, p, L/ν,M), covering A1; note that at this stage bothK1 and K2 depend on n,N, p, L/ν,M . Summing up the resulting inequalities, thatis (8.58) for Q ≡ Qi, we have

(8.59)∫

A1

|Du|q dx ≤ c(∫

A1

(1 + |Du|)p dx

) qp

+ c

∫

A1

(1 + |Du0|)p dx ,

again with c ≡ c(n,N, p, L/ν,M). At this point the proof of (8.55) follows byobserving that testing (8.53) by u− u0 and using (4.12)1 it follows that

(∫

A1

|Du|p dx) q

p

≤ c(∫

A1

(1 + |Du0|)p dx

) qp

≤ c∫

A1

(1 + |Du0|)p dx

where in the last estimate we used Holder’s inequality. Combining this last estimatewith (8.59) yields (8.55) for AR ≡ A1; the proof of the Proposition in the case of adomain of the type AR is therefore concluded. The proof in the case the underlyingintegration domain is a ball BR ≡ AR is exactly the same one. ¤

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