New results on Γ-limits of integral functionals

Ann. I. H. Poincaré – AN 31 (2014) 185–202

www.elsevier.com/locate/anihpc

New results on Γ -limits of integral functionals

Nadia Ansini

, Gianni Dal Maso

, Caterina Ida Zeppieri

aDipartimento di Matematica “G. Castelnuovo”, Sapienza Università di Roma, Piazzale Aldo Moro 2, 00185 Roma, Italy bSISSA, Via Bonomea 265, 34136 Trieste, Italy

cInstitut für Numerische und Angewandte Mathematik, Universität Münster, Einsteinstr. 62, 48149 Münster, Germany Received 7 March 2012; received in revised form 15 February 2013; accepted 21 February 2013

Available online 1 March 2013

Abstract

For ψ ∈ W^1,p(Ω;R^m) and g ∈ W^−1,p(Ω;R^d), 1 < p < +∞, we consider a sequence of integral functionals F_k^ψ,g:W^1,p(Ω;R^m)×L^p(Ω;R^d×n)→ [0,+∞]of the form

F_k^ψ,g(u, v)= ^Ωf_k(x,∇u, v) dx ifu−ψ∈W₀^1,p(Ω;R^m)and divv=g,

+∞ otherwise,

where the integrandsf_k satisfy growth conditions of orderp, uniformly ink. We prove aΓ-compactness result forF_k^ψ,gwith respect to the weak topology ofW^1,p(Ω;R^m)×L^p(Ω;R^d×n)and we show that under suitable assumptions the integrand of the Γ-limit is continuously differentiable. We also provide a result concerning the convergence of momenta for minimizers ofF_k^ψ,g.

©2013 Elsevier Masson SAS. All rights reserved.

Résumé

Pour tout ψ∈W^1,p(Ω;R^m)etg∈W^−1,p(Ω;R^d), 1< p <+∞, nous considérons une suite de fonctionnelles intégrales F_k^ψ,g:W^1,p(Ω;R^m)×L^p(Ω;R^d×n)→ [0,+∞]définies par

F_k^ψ,g(u, v)= ^Ωf_k(x,∇u, v) dx siu−ψ∈W₀^1,p(Ω;R^m)et divv=g,

+∞ sinon,

où les intégrandes f_k satisfont des conditions de croissance d’ordre p, uniformément en k. Nous démontrons un résultat de Γ-compacité pourF_k^ψ,gpar rapport à la topologie faible surW^1,p(Ω;R^m)×L^p(Ω;R^d×n)et nous prouvons que sous des condi- tions appropriées, l’intégrande de laΓ-limite est continûment différentiable. Nous montrons également un résultat de convergence des moments pour les minima deF_k^ψ,g.

©2013 Elsevier Masson SAS. All rights reserved.

MSC:35D99; 35E99; 49J45

Keywords:Γ-convergence; Integral functionals; Localization method;(curl,div)-quasiconvexity; Convergence of minimizers; Convergence of momenta

* Corresponding author.

E-mail addresses:ansini@mat.uniroma1.it(N. Ansini),dalmaso@sissa.it(G. Dal Maso),caterina.zeppieri@uni-muenster.de(C.I. Zeppieri).

0294-1449/$ – see front matter ©2013 Elsevier Masson SAS. All rights reserved.

http://dx.doi.org/10.1016/j.anihpc.2013.02.005

1. Introduction

In this paper we consider sequences of integral functionalsF_k:W^1,p(Ω;R^m)×L^p(Ω;R^d×n)→ [0,+∞)of the form

F_k(u, v)=

Ω

f_k(x,∇u, v) dx, (1.1)

whereΩ is a bounded open set inRⁿ and the integrandsfk satisfy suitable coerciveness and growth conditions of orderp∈(1,∞), uniformly ink(see (2.1) below). Specifically we are interested in the asymptotic behavior of the solutions to the following minimization problems

min

Fk(u, v): u−ψ∈W₀^1,p Ω;R^m

, v∈L^p

Ω;R^d×n

,divv=g

, (1.2)

whereψ∈W^1,p(Ω;R^m)andg∈W^−1,p(Ω;R^d)are given. The relevance of this setting where the functionals are defined on pairs(u, v)satisfying the differential constraint(curlu,divv)=(0, g)lies in the fact that in many appli- cations (see e.g. the case of electromagnetism) PDEs constraints of this type arise naturally.

To take into account the boundary and divergence constraints onuandv, respectively, we introduce the functionals F_k^ψ,g(u, v)=

F_k(u, v) ifu−ψ∈W₀^1,p(Ω;R^m)and divv=g,

+∞ otherwise.

(1.3) The main result of the present paper is as follows: if fk is a sequence of functions satisfying (2.1) and Fk are as in (1.1), there exist a subsequence offk, not relabeled, and a functionf such that the sequenceF_k^ψ,g Γ-converges to the corresponding functionalF^ψ,g, with respect to the weak topology ofW^1,p(Ω;R^m)×L^p(Ω;R^d×n). Moreover, the integrandfdoes not depend onψandg(see Theorem2.1and Theorem3.3). The case of functionals independent ofu, with the constraint divv=0, has been studied in[2], while similar problems in the framework ofA-quasiconvexity have been studied in[9]. However, it does not seem that the techniques used in these papers can lead directly, in our case, to a limit integrandf independent ofg.

We prove our main result in a nonconstructive way, following the so-called localization method ofΓ-convergence.

To this end, for every open setU⊆Ωwe consider the functionals F_k(u, v, U )=

U

f_k(x,∇u, v) dx. (1.4)

Notice that at this first stage both the boundary conditionu=ψand the constraint divv=gare omitted. In order to add the divergence constraint in the final step of the proof, it is convenient to introduce the following distance

d

(u₁, v₁), (u₂, v₂)

:= u₁−u₂L^p(Ω;R^m)+ v₁−v₂W^−1,p(Ω;R^d×n)+ div(v1−v₂)

W^−1,p(Ω;R^d) (1.5) for which(W^1,p(Ω;R^m)×L^p(Ω;R^d×n), d)is separable. Then, thanks to the general theory ofΓ-convergence in separable metric spaces, in Section2we prove that there exist a subsequence off_k, not relabeled, and a functionfsuch that for every open setU⊆Ω the functionals F_k(·,·, U ) Γ (d)-converge to the functionalF (·,·, U )corresponding tof (see Theorem2.3). In particular this gives theΓ (d)-convergence forU=Ω(see Theorem2.1). We also prove that under suitable assumptions on f_k, for a.e. x∈Ω the integrandf (x,·,·)of the Γ (d)-limit F is continuously differentiable (see Theorem2.8).

By virtue of the above results, in Section 3we deduce that the functionals F_k^ψ,g Γ-converge toF^ψ,g with re- spect to the weak topology ofW^1,p(Ω;R^m)×L^p(Ω;R^d×n). By general properties ofΓ-convergence this gives the convergence of minima and minimizers.

In Section4we also prove a result about the convergence of momenta for minimizers. Specifically, in Corollary4.6 we show that, if(u_k, v_k)is a minimizer ofF_k^ψ,g, then there exist a subsequence of(u_k, v_k), not relabeled, and a mini- mum point(u, v)ofF^ψ,g such thatu_k uweakly inW^1,p(Ω;R^m),v_k vweakly inL^p(Ω;R^d×n)(convergence of minimizers), and

∂_ξf_k(x,∇u_k, v_k) ∂_ξf (x,∇u, v) weakly inL^q

Ω;R^m×n

, (1.6)

∂_ηf_k(x,∇u_k, v_k) ∂_ηf (x,∇u, v) weakly inL^q

Ω;R^d×n

(1.7) (convergence of momenta). More in general we show that (1.6) and (1.7) can be obtained without assuming the minimality of(u_k, v_k)(see Theorem4.5). Indeed, the only hypotheses we need are:u_k uweakly inW^1,p(Ω;R^m), v_k vweakly inL^p(Ω;R^d×n), and

Ω

f_k(x,∇u_k, v_k) dx→

Ω

f (x,∇u, v) dx.

We finally remark that in the recent paper[3], inspired by previous results contained in[10,16], it is shown that minimum problems like (1.2) naturally arise when dealing with sequences of Dirichlet problems of the type

−div(σk∇w_k)=h inΩ, w_k∈H₀¹(Ω),

with(σk)uniformly elliptic and non-symmetric. In this respect, the results contained in the present paper are used in[1]to provide aΓ-convergence approach to the study ofH-convergence of non-symmetric linear elliptic operators (see also[3]). Namely, thanks to Proposition2.6and Theorem4.5of the present paper, in[1]we give an alternative and purely variational proof of the compactness ofH-convergence, originally proved by other methods by Murat and Tartar[17,18].

2. Γ-convergence of integral functionals

In this section we prove a compactness result, with respect toΓ-convergence, for integral functionals depending on the gradient of a vector-valued function and on a matrix-valued field.

LetΩ be a bounded open set inRⁿand let 1< p <+∞. Letf_k:Ω×R^m×n×R^d×n→ [0,+∞)be a sequence of Borel functions satisfying the following growth conditions of orderp: there exista0, a1>0 and two nonnegative functionsb0, b1∈L¹(Ω)such that for almost everyx∈Ω

a₀

|ξ|^p+ |η|^p

−b₀(x)f_k(x, ξ, η)a₁

|ξ|^p+ |η|^p

+b₁(x), (2.1)

for everyk∈N,ξ ∈R^m×n, andη∈R^d×n.

Consider the sequence of integral functionalsF_k:W^1,p(Ω;R^m)×L^p(Ω;R^d×n)→ [0,+∞)defined as follows F_k(u, v):=

Ω

f_k(x,∇u, v) dx. (2.2)

OnW^1,p(Ω;R^m)×L^p(Ω;R^d×n)consider the distanced defined by (1.5). Clearly we have

d

(u_k, v_k), (u, v)

→0 ⇔

⎧⎪

⎨

⎪⎩

u_k→u strongly inL^p Ω;R^m

, v_k→v strongly inW^−1,p

Ω;R^d×n , divv_k→divv strongly inW^−1,p

Ω;R^d . Notice that(W^1,p(Ω;R^m)×L^p(Ω;R^d×n), d)is a separable metric space.

The following compactness theorem is the main result of this section.

Theorem 2.1(Γ-compactness of integral functionals). LetF_kbe the sequence of functionals defined in(2.2)withf_k satisfying(2.1). Then, there exist a subsequenceF_kj and a Borel functionf:Ω×R^m×n×R^d×n→ [0,+∞)such that the functionalsF_kj Γ (d)-converge to the functionalF:W^1,p(Ω;R^m)×L^p(Ω;R^d×n)→ [0,+∞)defined as

F (u, v):=

Ω

f (x,∇u, v) dx, (2.3)

withf satisfying

a₀

|ξ|^p+ |η|^p

−b₀(x)f (x, ξ, η)a₁

|ξ|^p+ |η|^p

+b₁(x), (2.4)

f (x, ξ₁, η₁)−f (x, ξ₂, η₂)a₂

|ξ₁−ξ₂| + |η₁−η₂|

|ξ₁| + |ξ₂| + |η₁| + |η₂| +b₂(x)p−1

(2.5) for almost every x∈Ω, ξ, ξ₁, ξ₂∈R^m×n, η, η₁, η₂∈R^d×n, where a₂∈R⁺ and b₂∈L^p(Ω)⁺ depend only on a₀, a₁, b₀, b₁.

We prove Theorem2.1in a nonconstructive way, following the so-called localization method ofΓ-convergence, for which we refer the reader to[12, Chapters 14–20]. Loosely speaking, this method consists of two main steps. In the first one, based on compactness arguments, we prove the existence of aΓ-converging (sub)sequence. While in the second one we recover enough information on the structure of theΓ-limit as to obtain a representation in an integral form.

As a preliminary step, we localize the sequenceFkby introducing an explicit dependence on the set of integration.

With a little abuse of notation, we consider the functionals F_k(u, v, B):=

B

f_k(x,∇u, v) dx, (2.6)

defined for every u∈W^1,p(Ω;R^m), v ∈L^p(Ω;R^d×n), and for every Borel set B⊆Ω, so that F_k(u, v, Ω)= F_k(u, v).

LetA(Ω)be the set of all open subset ofΩ. The following compactness theorem shows that there exists a sub- sequence whoseΓ-limitF (u, v, U )is a measure with respect toU. This is a preliminary step to obtain the integral representation (2.8) below.

Proposition 2.2(Γ-compactness). LetF_k:W^1,p(Ω;R^m)×L^p(Ω;R^d×n)×A(Ω)→ [0,+∞)be the sequence of functionals defined by(2.6), withf_ksatisfying the growth conditions(2.1). Then, there exist a subsequenceF_kj and a local functionalF:W^1,p(Ω;R^m)×L^p(Ω;R^d×n)×A(Ω)→ [0,+∞)such that

F_kj(·,·, U ) Γ (d)-converges toF (·,·, U )

for every U∈A(Ω). Moreover, for all(u, v)∈W^1,p(Ω;R^m)×L^p(Ω;R^d×n)the set functionF (u, v,·)is the re- striction toA(Ω)of a nonnegative Borel measure defined onΩ. Finally,

F (u+s, v, U )=F (u, v, U ) (2.7)

for every(u, v)∈W^1,p(Ω;R^m)×L^p(Ω;R^d×n),s∈R^m, andU∈A(Ω).

Proof. Arguing as in[12, Theorem 19.1]we can prove that the sequenceF_k satisfies the fundamental estimate intro- duced in[12, Definition 18.2]. Since(W^1,p(Ω;R^m)×L^p(Ω;R^d×n), d)is a separable metric space, the result can be obtained by adapting the arguments of[12, Section 18]. The functionalF is local by[12, Proposition 16.15]. The final statement of the proposition is trivial. 2

Using Proposition2.2, we now provide an integral representation formula for theΓ (d)-limitF. The proof relies on standard arguments that we repeat and adapt to our context for the reader’s convenience. Note that Theorem2.1 follows from Theorem2.3below by takingU=Ω.

Theorem 2.3(Γ-compactness of local integral functionals). Letfk:Ω×R^m×n×R^d×n→ [0,+∞)be Borel functions satisfying the growth assumptions(2.1), and letFk be as in(2.6). Then, there exist a subsequenceFk_j and a Borel functionf:Ω×R^m×n×R^d×n→ [0,+∞), satisfying(2.4)and(2.5), such that for everyU∈A(Ω)the functionals F_kj(·,·, U ) Γ (d)-converge to the functionalF (·,·, U ):W^1,p(Ω;R^m)×L^p(Ω;R^d×n)→ [0,+∞)defined as

F (u, v, U ):=

U

f (x,∇u, v) dx. (2.8)

Proof. Proposition 2.2 ensures the existence of a subsequence F_kj(·,·, U ) that Γ (d)-converges to a functional F (·,·, U ), for everyU∈A(Ω). Hence, it remains to deduce the integral representation formula (2.8). This is done in several steps.

Step 1. Definition off.

Fix (ξ, η)∈R^m×n×R^d×n and define u_ξ(x)=ξ x; by Proposition2.2 F (u_ξ, η,·) can be extended to a Borel measure onΩ which, by (2.1), is absolutely continuous with respect to the Lebesgue measure. For everyx∈Ω we define

f (x, ξ, η):=lim sup

ρ→0⁺

F (uξ, η, Bρ(x))

|Bρ(x)| , (2.9)

whereB_ρ(x)is then-dimensional ball of radiusρ >0, centered atx. Thenf is a Borel function and by the Lebesgue Differentiation Theorem we have

F (u_ξ, η, U )=

U

f (x, ξ, η) dx, (2.10)

for everyU∈A(Ω). By (2.1) it follows that (2.4) holds at every Lebesgue point common tob0andb1. Step 2. Integral representation on piecewise affine and piecewise constant functions.

SinceF is local andF (u, v,·)is a measure, from (2.7) and (2.10) we obtain that F (u, v, U )=

U

f (x,∇u, v) dx

whenuis piecewise affine andvis piecewise constant (we assume that the boundaries of the sets whereuis affine andvis constant have zero Lebesgue measure). We refer to the proof of[12, Theorem 20.1]for the details.

Step 3. Convexity properties off.

The proof of the rank-1-convexity inξ is standard (see, e.g.,[8, Theorem 9.1]). Let us prove thatf is rank-(n−1) convex inη,i.e., for everyt∈(0,1)

f

x, ξ, tη₁+(1−t )η₂

tf (x, ξ, η₁)+(1−t )f (x, ξ, η₂) (2.11)

for everyx∈Ω, for everyξ∈R^m×n, and for everyη₁, η₂∈R^d×nwith rank(η1−η₂)n−1. By (2.9) it suffices to show that, ifB_ρ(x)⊂Ω, then for allt∈(0,1)

F

u_ξ, t η₁+(1−t )η₂, B_ρ(x) t F

u_ξ, η₁, B_ρ(x)

+(1−t )F

u_ξ, η₂, B_ρ(x)

for everyξ∈R^m×nand for everyη₁, η₂∈R^d×nwith rank(η1−η₂)n−1. This inequality can be obtained as in[2, Theorem 4.2, Step 3]. Indeed, as a consequence of the rank property ofη1, η2we deduce the existence of a unit vector ν∈Sⁿ⁻¹ such that(η1−η2)ν=0. Then, if we definev:Rⁿ→ {η1, η2}as v(y)=η1ify ∈U₁^ν andv(y)=η2if y∈U₂^ν, with

U₁^ν:=

y∈Rⁿ: hy·ν < h+t, h∈Z , U₂^ν:=

y∈Rⁿ: h+ty·ν < h+1, h∈Z ,

we clearly have divv=0. These sets represent a lamination ofRⁿ in the direction orthogonal to ν, with volume fractiont and 1−t. If we definevh(y):=v(hy)fory∈Rⁿ, it is easy to show that

v_h tη1+(1−t )η2 weakly^∗inL^∞

Ω;R^d×n

,ash→ ∞.

Hence, in particularvhconverges strongly tovinW^−1,p(Ω;R^d×n)and divvh=0=divv.

Moreover, settingU_1,h^ν :=(1/ h)U₁^ν andU_2,h^ν :=(1/ h)U₂^ν, we have χ_U^ν

1,h t and χ_U^ν

2,h1−t weakly^∗inL^∞(Ω),ash→ ∞.

By Step 2 and by the lower semicontinuity ofF with respect to the metricd, we have F

u_ξ, t η₁+(1−t )η₂, B_ρ(x)

lim inf

h→∞ F

u_ξ, v_h, B_ρ(x)

=lim inf

h→∞

U_1,h^ν ∩Bρ(x)

f (y, ξ, η1) dy+

U_2,h^ν ∩Bρ(x)

f (y, ξ, η2) dy

=t

B_ρ(x)

f (y, ξ, η1) dy+(1−t )

B_ρ(x)

f (y, ξ, η2) dy

=t F

uξ, η1, Bρ(x)

+(1−t )F

uξ, η2, Bρ(x) , for everyξ∈R^m×n. Finally, taking the limsup inρyields (2.11).

The rank-1 convexity in ξ together with the rank-(n−1)convexity inη ensures thatf is separately convex in each component;i.e., ifξ =(ξ )_ij andη=(η)_j,f is convex inξ_ij for everyi=1, . . . , mandj=1, . . . , n, and it is convex inη_jfor every=1, . . . , dandj=1, . . . , n. Therefore, the growth condition (2.4) together with the separate convexity yields thatf (x,·,·)is locally Lipschitz (see, e.g.,[11, Lemma 2.2]). More precisely, there exista2>0 and a nonnegative functionb2∈L^p(Ω)such that (2.5) holds.

Step 4. Integral representation.

By (2.4) and (2.5) for everyU∈A(Ω)the functional (u, v)→

U

f (x,∇u, v) dx (2.12)

is continuous with respect to the strong convergence ofW^1,p(Ω;R^m)×L^p(Ω;R^d×n).

Let(u, v)∈W^1,p(Ω;R^m)×L^p(Ω;R^d×n)and letU∈A(Ω)withU⊂⊂Ω. We can find a sequence of functions u_h∈W^1,p(Ω;R^m)strongly converging touinW^1,p(Ω;R^m)with piecewise affine restrictions toUand a sequence of piecewise constant functions vh strongly converging tovinL^p(Ω;R^d×n). Note that(uh, vh)converge to(u, v) with respect to the distanced. SinceF is lower semicontinuous and (2.12) is continuous, by Step 2 we get

F (u, v, U )lim inf

h→∞ F (u_h, v_h, U )= lim

h→∞

U

f (x,∇u_h, v_h) dx=

U

f (x,∇u, v) dx.

We now prove the converse inequality. Let (u, v)∈W^1,p(Ω;R^m)×L^p(Ω;R^d×n) and let G:W^1,p(Ω;R^m)× L^p(Ω;R^d×n)×A(Ω)→ [0,+∞)be defined by

G(u,˜ v, U )˜ :=F (u+ ˜u, v+ ˜v, U ).

SinceGsatisfies the same properties asF, there exists a Carathéodory functiong:Ω×R^m×n×R^d×n→ [0,+∞) such that for every(u,˜ v)˜ ∈W^1,p(Ω;R^m)×L^p(Ω;R^d×n)and everyU∈A(Ω), withU⊂⊂Ω, we have that

G(u,˜ v, U )˜

U

g(x,∇ ˜u,v) dx,˜ (2.13)

with equality wheneveru˜andv˜are piecewise affine and piecewise constant, respectively.

Let(u_h, v_h)be the approximating functions considered above; then

U

g(x,0,0) dx=G(0,0, U )=F (u, v, U )

U

f (x,∇u, v) dx

= lim

h→∞

U

f (x,∇uh, vh) dx= lim

h→∞F (uh, vh, U )= lim

h→∞G(uh−u, vh−v, U )

lim

h→∞

U

g

x,∇(u_h−u), v_h−v dx=

U

g(x,0,0) dx.

Hence all inequalities are actually equalities, and in particular F (u, v, U )=

U

f (x,∇u, v) dx, (2.14)

for all(u, v)∈W^1,p(Ω;R^m)×L^p(Ω;R^d×n)andU∈A(Ω), withU⊂⊂Ω. Finally, the above equality holds for all U∈A(Ω)sinceF is the restriction toA(Ω)of a Borel measure. 2

Remark 2.4.The functionalF is lower semicontinuous with respect to the distanced defined by (1.5), then we can apply[14, Theorem 3.6]to obtain that for a.e.x∈Ωthe function(ξ, η)→f (x, ξ, η)isA-quasiconvex according to [14, Definition 3.1]where

A(ψ, v):=(curlψ,divv)

for every(ψ, v)∈L^p(Ω;R^m×n×R^d×n). More precisely, we say thatf (x,·,·)is(curl,div)-quasiconvex if f (x, ξ, η)

Q

f

x, ξ+ψ (y), η+v(y) dy

for all(ξ, η)∈R^m×n×R^d×nand all(ψ, v)∈C^∞(Rⁿ;R^m×n×R^d×n),Q-periodic, such that curlψ=0, divv=0,

Qψ dy=0,

Qv dy=0. In view of (2.4) and (2.5) in the above definition we may replaceC^∞(Rⁿ;R^m×n×R^d×n) byL^p(Rⁿ;R^m×n×R^d×n)(see[14, Remark 3.3]).

We consider now the special case of quadratic functionals depending on the gradient of a scalar functionuand on a vector fieldv. In Corollary2.5and Proposition2.6we choosem=d=1 andp=2.

Corollary 2.5.LetΣk∈L^∞(Ω;R^2n×2n)be symmetric matrices such that for a.e.x∈Ω, and for everyw∈R²ⁿ

c₀|w|²Σ_k(x)w·wc₁|w|² (2.15)

for some0< c₀c₁independent ofk, where the dot denotes the scalar product. LetQ_k:W^1,2(Ω)×L²(Ω;Rⁿ)× A(Ω)→ [0,+∞)be the sequence of quadratic forms defined as follows

Qk(u, v, U ):=

U

Σk(x) ∇u

v

· ∇u

v

dx. (2.16)

Then, there exist a subsequenceQk_j and a matrixΣ∈L^∞(Ω;R^2n×2n)satisfying(2.15), such that for everyU∈ A(Ω)the quadratic formsQ_kj(·,·, U ) Γ (d)-converge to the quadratic formQ(·,·, U ):W^1,2(Ω)×L²(Ω;Rⁿ)→ [0,+∞)given by

Q(u, v, U ):=

U

Σ (x) ∇u

v

· ∇u

v

dx. (2.17)

Proof. By Theorem2.3and[12, Theorem 11.10]we have that for everyU∈A(Ω)the quadratic formsQk_j(·,·, U ) Γ (d)-converge to the quadratic formQ(·,·, U )given by

Q(u, v, U )=

U

f (x,∇u, v) dx.

By (2.9) and[12, Proposition 11.9]on the characterization of the quadratic forms, we get that for everyx∈Ω also f (x,·,·):Rⁿ×Rⁿ→ [0,+∞)is a quadratic form. Therefore, there exists a matrixΣ∈L^∞(Ω;R^2n×2n)satisfying (2.15) such that

f (x, ξ, η)=Σ (x) ξ

η

· ξ

η

, for every(x, ξ, η)∈Ω×Rⁿ×Rⁿ. 2

For future applications (see, e.g.,[1]) it is useful to highlight the following fact.

Proposition 2.6.LetΣk, Σ∈L^∞(Ω;R^2n×2n)be symmetric matrices satisfying conditions(2.15). LetQk(u, v, U ) and Q(u, v, U ) be quadratic forms defined as in (2.16), (2.17), and let Q_k(u, v):=Q_k(u, v, Ω), Q(u, v):=

Q(u, v, Ω). IfQ_k Γ (d)-converges toQthen for everyU ∈A(Ω)the quadratic formsQ_k(·,·, U ) Γ (d)-converge to the quadratic formQ(·,·, U ).

Proof. Following the argument of[3, Theorem 4.6]we may prove that the matrixΣis independent of the setU. 2 We now prove that the convergence in measure of the integrandsf_ktogether with(curl,div)-quasiconvexity implies Γ (d)-convergence of the corresponding integral functionalsF_k.

Theorem 2.7.Letf_kbe a sequence of Borel functions satisfying(2.1)and such thatf_k(x,·,·)is(curl,div)-quasiconvex for a.e.x∈Ω. Assume that

fk(·, ξ, η)→f (·, ξ, η) in measure onΩ (2.18)

for everyξ∈R^m×nandη∈R^d×n. Thenf is a Borel function, it satisfies(2.4), andf (x,·,·)is(curl,div)-quasiconvex for a.e.x∈Ω. LetF_kandF be the functionals defined by(2.2)and(2.3). ThenF_k Γ (d)-converges toF.

Proof. The first statement is a straightforward consequence of (2.18). Sincefk(x,·,·)andf (x,·,·)are(curl,div)- quasiconvex, by[14, Proposition 3.4]f_k(x,·, η)andf (x,·, η)are rank-1 convex onR^m×nfor a.e.x∈Ω and every η∈R^d×n, whilef_k(x, ξ,·)andf (x, ξ,·)are rank-(n−1)convex onR^d×nfor a.e.x∈Ωand everyξ∈R^m×n. Hence, they are separately convex with respect to the scalar components ofξ andη. This property, together with the growth assumptions (2.1) and (2.4), implies thatf_k andf satisfy the continuity estimates (2.5) uniformly with respect tok (see also Step 3 of the proof of Theorem2.3).

Under our hypotheses we can prove that there exists a subsequence off_k, not relabeled, such thatf_k(x, ξ, η)→ f (x, ξ, η)for a.e.x∈Ω, for everyξ ∈Q^m×n,η∈Q^d×n whereQdenotes the set of rational numbers. By (2.5) and a diagonal argument the same property holds for a.e. x∈Ω, for every ξ ∈R^m×n,η∈R^d×n. Taking into account the growth conditions (2.1), this implies thatF_k(u, v)→F (u, v)for every(u, v)∈W^1,p(Ω;R^m)×L^p(Ω;R^d×n).

Therefore to achieve theΓ-convergence result it is enough to prove theΓ-liminf inequality. To this end let(u, v)∈ W^1,p(Ω;R^m)×L^p(Ω;R^d×n)and let(u_k, v_k)∈W^1,p(Ω;R^m)×L^p(Ω;R^d×n)be a sequence such that(u_k, v_k)→ (u, v)in the distancedand such thatF_k(u_k, v_k)converges to a finite number. We want to prove that

F (u, v) lim

k→∞Fk(uk, vk). (2.19)

From the convergence in the distance d and the boundedness of F_k(u_k, v_k) we deduce that u_k u weakly in W^1,p(Ω;R^m)andv_k v weakly inL^p(Ω;R^d×n). Up to subsequence we may assume that (∇u_k, v_k)generates a Young measureν:Ω→M(R^m×n×R^d×n). We refer to[14, Section 2]for the properties of Young measures we use in this proof. For a general treatment of this subject we refer to[5,7,18].

We now prove that

Ω

R^m×n×R^d×n

f (x, ξ, η) dν_x(ξ, η)

dx lim

k→∞

Ω

f_k(x,∇u_k, v_k) dx. (2.20)

Arguing as in[12, Theorem 5.14]for everyε >0 we find a measurable setA_ε⊂Ω, with|A_ε|< ε, such that for every M >0 we have

f_k(x, ξ, η)→f (x, ξ, η) uniformly on(Ω\A_ε)×B_M, (2.21) whereB_M is the ball inR^m×n×R^d×n with center(0,0)and radiusM. We can easily construct a sequenceϕ_M of Carathéodory functions such that

ϕ_M(x, ξ, η)→f (x, ξ, η) asM→ +∞for a.e.x∈Ωand for every(ξ, η)∈R^m×n×R^d×n, (2.22a) ϕ_M(x, ξ, η)=0 for a.e.x∈Ω and for every(ξ, η) /∈B_M, (2.22b) 0ϕ_M(x, ξ, η)

f (x, ξ, η)− 1 M

₊

for a.e.x∈Ωand for every(ξ, η)∈R^m×n×R^d×n, (2.22c) where(t)⁺denotes the positive part oft∈R. By (2.21) and (2.22) there existsk_M∈Nsuch that for everykk_M we haveϕ_M(x, ξ, η)f_k(x, ξ, η)for a.e.x∈Ω\A_εand for every(ξ, η)∈R^m×n×R^d×n. Integrating this inequality we obtain

Ω\A_ε

ϕM(x,∇uk, vk) dx

Ω

fk(x,∇uk, vk) dx (2.23)

for everykk_M. By the Fundamental Theorem on Young Measures (see e.g.[14, Theorem 2.2])

Ω\Aε

R^m×n×R^d×n

ϕ_M(x, ξ, η) dν_x(ξ, η)

dxlim inf

k→∞

Ω\Aε

ϕ_M(x,∇u_k, v_k) dx.

Taking into account (2.23) and passing to the limit first asM→ ∞and then as ε→0, thanks to (2.22a) we ob- tain (2.20).

To prove (2.19) it remains to show that f

x,∇u(x), v(x)

R^m×n×R^d×n

f (x, ξ, η) dν_x(ξ, η)

for a.e. x∈Ω. Since f (x,·,·)is (curl,div)-quasiconvex, this can be done arguing as in the proof of[14, Theo- rem 3.7]. This concludes the proof of theΓ-liminf inequality and therefore shows that the selected subsequence ofF_k Γ (d)-converges toF. Finally, since theΓ-limitF does not depend on the subsequence, we may conclude thanks to the Urysohn property ofΓ-convergence[12, Proposition 8.3]. 2

We now prove that under suitable assumptions the integrandf of theΓ (d)-limitF is continuously differentiable with respect to (ξ, η). Similar results have been proved in [15]for the convex case and in [4,6] for quasiconvex envelopes.

Theorem 2.8.Letf_k:Ω×R^m×n×R^d×n→ [0,+∞)be Borel functions satisfying(2.1). Assume that for everyk, for a.e.x∈Ω, and for everyξ₁, ξ₂∈R^m×n,η₁, η₂∈R^d×n

(ξ, η)→fk(x, ξ, η) belongs toC¹

R^m×n×R^d×n

, (2.24a)

∂_ξf_k(x, ξ₁, η)−∂_ξf_k(x, ξ₂, η)a|ξ₁−ξ₂|^α

|ξ₁| + |ξ₂| + |η| +b(x)p−1−α

, (2.24b)

∂_ηf_k(x, ξ, η₁)−∂_ηf_k(x, ξ, η₂)a|η₁−η₂|^α

|ξ| + |η₁| + |η₂| +b(x)p−1−α

, (2.24c)

wherea >0,b∈L^p(Ω)⁺, and0< α <min{1, (p−1)}are independent ofk.

Let f :Ω ×R^m×n×R^d×n→ [0,+∞) be a Borel functions satisfying(2.4), (2.5), and let Fk and F be the functionals defined by(2.6)and(2.8). Assume that for everyU∈A(Ω)

F_k(·,·, U ) Γ (d)-converges toF (·,·, U ). (2.25)

Then,

(ξ, η)→f (x, ξ, η) belongs toC¹

R^m×n×R^d×n

(2.26) for a.e.x∈Ω.

Proof. Givenx∈Ωand 0< ρ <dist(x, ∂Ω)we define F_k^x,ρ(u, v):=

B₁

f_k^x,ρ

y,∇u(y), v(y)

dy, F^x,ρ(u, v):=

B₁

f^x,ρ

y,∇u(y), v(y) dy,

whereB₁is the open ball inRⁿwith center 0 and radius 1 and

f_k^x,ρ(y, ξ, η):=f_k(x+ρy, ξ, η), f^x,ρ(y, ξ, η):=f (x+ρy, ξ, η).

By (2.25) and by a change of variables we obtain thatF_k^x,ρ Γ (d)-converges toF^x,ρ, where now the distanced is defined usingB₁instead ofΩ. Thanks to the continuity assumptions (2.5) there exists a negligible setN⊂Ω such that everyx∈Ω\N is a Lebesgue point off (·, ξ, η)for every(ξ, η)∈R^m×n×R^d×n, as well a Lebesgue point of the functionb0andb1in (2.4). It follows that for everyx∈Ω\N,f^x,ρ(·, ξ, η)→f (x, ξ, η)in measure onB1as ρ→0. Moreover, there exist a sequenceρi→0, possibly depending onx, and two functionsb₀^xandb^x₁such that

b₀(x+ρ_iy)b^x₀(y), b₁(x+ρ_iy)b₁^x(y)

for everyiand for a.e.y∈B₁. By Remark2.4for a.e.x∈Ω the functionsf^x,ρi(y,·,·)are(curl,div)-quasiconvex for a.e.y∈B₁. Therefore for everyx∈Ω\N we can apply Theorem2.7and obtain that

F^x,ρi Γ (d)-converges toF^x asρ_i→0, where

F^x(u, v):=

B1

f

x,∇u(y), v(y) dy.

By diagonal argument we find a sequencek_i such thatF_k^x,ρi

i Γ (d)-converges toF^x.

We can now apply the arguments of the proof of[15, Proposition 2.5]using Lemma4.4in place of[15, Lemma 2.4]

and we obtain the existence of the partial derivatives off with respect to the components of(ξ, η). The continuity with respect to(ξ, η)of these partial derivatives is a consequence of the convexity in each components which, in its turn, follows from the(curl,div)-quasiconvexity. 2

3. Boundary condition and divergence constraint

In this section we study functionals with a Dirichlet boundary conditionu=ψand divergence constraint divv=g.

We begin with the case of boundary conditions.

Theorem 3.1 (Γ-convergence with boundary data). Let Fk be the sequence of functionals defined in (2.2) with f_k satisfying (2.1). Assume that F_k Γ (d)-converges to F satisfying (2.3)–(2.5). Let ψ∈W^1,p(Ω;R^m) and let F_k^ψ:W^1,p(Ω;R^m)×L^p(Ω;R^d×n)→ [0,+∞]be the functionals defined as

F_k^ψ(u, v):=

F_k(u, v) ifu−ψ∈W₀^1,p(Ω;R^m),

+∞ otherwise. (3.1)

Then, the functionalsF_k^ψΓ (d)-converge to F^ψ(u, v):=

F (u, v) ifu−ψ∈W₀^1,p(Ω;R^m), +∞ otherwise.

Proof. Clearly, we have to deal only with theuvariable. Then, the proof exactly follows that of[12, Theorem 21.1].

Indeed, theΓ-liminf inequality is a straightforward consequence of (2.1), of the Poincaré inequality, and of the fact thatW₀^1,p(Ω;R^m)+ψis closed in the weak topology ofW^1,p(Ω;R^m). To prove theΓ-limsup inequality we have to exhibit a recovery sequence such thatu_k−ψ∈W₀^1,p(Ω;R^m), for everyk. To this end, we suitably modify a recovery sequence forFkas in[12, Theorem 21.1]. Then the fundamental estimate allows us to show that the error introduced in the energy goes to zero ask→ +∞. 2