Homogenization of convex functionals which are weakly coercive and not equi-bounded from above

Ann. I. H. Poincaré – AN 30 (2013) 547–571

www.elsevier.com/locate/anihpc

Homogenization of convex functionals which are weakly coercive and not equi-bounded from above

Marc Briane

, Juan Casado-Díaz

aInstitut de Recherche Mathématique de Rennes, INSA de Rennes, France bDpto. de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Spain

Received 17 February 2012; received in revised form 19 October 2012; accepted 23 October 2012 Available online 15 November 2012

Abstract

This paper deals with the homogenization of nonlinear convex energies defined inW₀^1,1(Ω), for a regular bounded open setΩ ofR^N, the densities of which are not equi-bounded from above, and which satisfy the following weak coercivity condition: There existsq > N−1 ifN >2, andq1 ifN=2, such that any sequence of bounded energy is compact inW₀^1,q(Ω). Under this assumption theΓ-convergence of the functionals for the strong topology ofL^∞(Ω)is proved to agree with theΓ-convergence for the strong topology ofL¹(Ω). This leads to an integral representation of theΓ-limit inC₀¹(Ω)thanks to a local convex density.

An example based on a thin cylinder with very low and very large energy densities, which concentrates to a line shows that the loss of the weak coercivity condition can induce nonlocal effects.

©2012 Elsevier Masson SAS. All rights reserved.

MSC:35B27; 35B50; 35J60

Keywords:Homogenization; Convex functionals; Nonlinear elliptic equations; Weak coercivity; Maximum principle

1. Introduction

Since the beginning of the seventies the homogenization theory has greatly developed through the G-convergence of operators[30], the H-convergence of PDE’s[29](see also[31]and the references therein), and theΓ-convergence of functionals[19,21](see also[18]for a review and the references therein). The De GiorgiΓ-convergence has been a powerful mathematical tool for studying the asymptotic behavior of minima of functionals defined for a regular bounded open setΩofR^N, by

F_n(v):=

Ω

f_n(x,∇v) dx, forv∈W₀^1,1(Ω). (1.1)

* Corresponding author.

E-mail addresses:[email protected](M. Briane),[email protected](J. Casado-Díaz).

0294-1449/$ – see front matter ©2012 Elsevier Masson SAS. All rights reserved.

http://dx.doi.org/10.1016/j.anihpc.2012.10.005

The seminal results in this sense were obtained in[20,15,17]. Assuming thatf_nis convex with respect to the second argument and satisfies the boundedness from above:

f_n(x, ξ )a_n(x)

1+ |ξ|^p

, for a.e.x∈Ω,∀ξ ∈R^N, ∀n∈N, (1.2)

for a fixedp >1 and for a given nonnegative bounded sequencea_n inL¹(Ω), anyΓ-limitF ofF_nfor the topology ofC₀⁰(Ω)was shown in[15,17]to have a similar integral representation, namely

F (v)=

Ω

f (x,∇v) dμ, forv∈C₀¹(Ω), (1.3)

wheref is convex with respect to the second argument andμis a Radon measure onΩ¯. Under the additional assump- tion of equi-integrability of the sequencea_nthe previous representation also holds for the strong topology ofL¹(Ω) as shown first in [20]. A few years later, it was proved in [22]that the loss of equi-integrability for equicoercive quadratic densitiesf_n may induce nonlocal effects in dimension three. A connection between this type of degener- acy and the Beurling–Deny [5]representation formula of the Dirichlet forms was established in[28]for quadratic functionals. Then, the closure set of the three-dimensional quadratic functionals with respect to theΓ-convergence for the strong topology ofL²(Ω)was obtained in[16]according to the Beurling–Deny theory. In the same spirit, the three-dimensional examples from[4]ofW^1,p(Ω)-equicoercive functionals forp >1, i.e.

∃C >0,∀n∈N, ∀v∈C_c¹(Ω), F_n(v)C

Ω

|∇v|^pdx, (1.4)

show no degeneracy of theirΓ-limits for the strong topology ofL^p(Ω)provided thatp >2, while nonlocal effects appear whenp∈(1,2], like in[22,4]. On the contrary, the case of dimension two with equicoercive functionals is quite different, since it was proved in [12–14]for quadratic functionals, and in [8]for W^1,p-equicoercive,p >1, convex periodic functionals, that theΓ-limits have a representation of type (1.3) in dimension two. On the other hand, the loss of coercivity may also induce degenerate limit behaviors in terms of coupled systems as shown for example in[24,28,25,9,7]. Also note that in periodic homogenization very weak coercivity conditions including perforated do- mains were treated using extension operators, weak notions of connectedness or multi-scale convergence approaches (see, e.g.,[1,2,10,32,33]). From a certain point of view all these works deal with the same question:

Under what conditions theΓ-limits of convex functionals of type(1.1)remain of type(1.3)?

The present work is an attempt to give a unified answer in any dimensionN 2, for sequences of convex func- tionalsF_nthe densities of which are neither equi-bounded from above nor equi-bounded from below. Our approach is based on the combination of three independent results:

• In Section2 we recover the result of[17](see Theorem2.4) but replacing theΓ-convergence for the topology ofC₀⁰(Ω)by theΓ-convergence for the strong topology ofL^∞(Ω). We also make an assumption on the convex densities fn, which is less restrictive than (1.2) (see conditions (2.11), (2.12), and Remark2.6), and needs an alternative approach.

• In Section3 we establish a general framework (see Corollary3.5) in which the Γ-convergence for the strong topology ofL^∞(Ω)agrees with theΓ-convergence for the strong topology ofL¹(Ω). This is the most original part of the paper. The strong equicoercivity condition (1.4) is now replaced by the following weaker condition:

There exists a real numberqwithq > N−1 ifN >2, andq=1 ifN=2, such that

⎧⎪

⎨

⎪⎩

∀n∈N,∀c >0, {F_nc}is sequentially compact inW₀^1,q(Ω)weak,

∀u_n∈W₀^1,1(Ω), lim sup

n→∞ F_n(u_n) <∞ ⇒ u_nconverges weakly inW₀^1,q(Ω), up to a subsequence,

(1.5) which holds for a large class of functionsf_n (see Proposition3.2). Under this condition we prove (see Theo- rem3.4) a uniform convergence for (roughly speaking) minimizersu_nofF_nwhich converge weakly inW₀^1,q(Ω) to a function inC⁰(Ω). The key ingredient is a maximum principle type result (see Lemma¯ 3.7) following the idea of[27](see also[23]), which allows us to deduce a uniform estimate foru_n from the compact embedding ofW^1,q(∂B)intoC⁰(∂B)for any ballB⊂Ω, due to the condition onq. The Aubin compactness theorem[3]

is also used in the caseN >2 andq > N−1, while it is replaced by the Kuratowski, Ryll-Nardzewski selection theorem[26]in the much more delicate caseN=2 andq=1.

• Section4is devoted to a counter-example, separating the casesN >2 (Theorem4.2) andN=2 (Theorem4.4), which shows that the weak coercivity condition (1.5) is actually crucial to obtain the local Γ-limit representa- tion (1.3). Indeed, the loss of condition (1.5) may induce nonlocal effects. The counter-example is based on a columnar structure like in[22,4]. But contrary to the three-dimensional periodic fiber reinforcement of[22,4], here the energy density f_ntakes both very low and very large values in one cylinder ifN >2, and one strip if N =2, which concentrates along a line asntends to∞. Based on the counter-example the importance of the weak coercivity condition (1.5) as well as the more precise conditions of Proposition3.2, for deriving a local Γ-limit is discussed in Remark4.1.

Therefore, the three previous results allow us to answer to the above question through the following:

Theorem 1.1. Let Ω be a bounded open set of R^N, N 2, with a Lipschitz boundary. Consider a sequence of nonnegative functionsfn:Ω×R^N→ [0,∞),n∈N, satisfying the properties(2.1)–(2.4), (2.11), (2.12)below. Also assume that the associated convex functionalFndefined by(1.1)satisfies the condition(1.5).

Then, there exist a subsequence ofn, still denoted byn, a Radon measureμonΩ, and a function¯ f :Ω×R^N→ [0,∞)satisfying the properties(2.13)–(2.16)below, such that the sequenceFnΓ-converges inC₀¹(Ω)for the strong topology ofL¹(Ω)to the functionalF defined by(1.3).

Focus on the particular two-dimensional case with quadratic densitiesf_n(x, ξ )=A_n(x)ξ·ξ, for(x, ξ )∈Ω×R², whereA_n is a sequence of positive definite symmetric matrix-valued functions defined onΩ. Then, Theorem1.1 and Proposition3.2forN=2 lead to a localΓ-limit of type (1.3) under the sole assumption that the inverse of the smallest eigenvalueλ_nof A_nis bounded and equi-integrable inL¹(Ω), without any prescribed bound from above.

Moreover, the two-dimensional counter-example of Section4 (see Theorem4.4) shows that the equi-integrability of λ⁻_n¹ inL¹(Ω)is actually essential. This extends the result of [14]obtained through an approach based on the Dirichlet forms, but for a sequenceλ_nwhich is bounded from below by a positive constant.

A few recalls and notations

We recall the definition of the De GiorgiΓ-convergence and some of its properties which will be used in the sequel.

We refer to[18]for an exhaustive presentation ofΓ-convergence (see also[6]for an elementary approach).

Definition 1.2.LetV be a metric space, and letF_n:V → [0,∞],n∈N, be a sequence of functionals. Forv∈V,F_n is said toΓ-converge toF (v)∈ [0,∞]atvif

i) theΓ-liminf inequality holds

∀vn→vinV , F (v)lim inf

n→∞ Fn(vn), (1.6)

ii) theΓ-limsup inequality holds

∃v_n→vinV , F (v)= lim

n→∞F_n(v_n). (1.7)

Any sequence satisfying (1.7) is called a recovery sequence forF_nof limitv.

LetW be a subset ofV. The sequenceF_n is said toΓ-converge inW toF :W → [0,∞]if for anyv∈W,F_n Γ-converges toF (v)atv.

Notations

• S_N−1denotes the unit sphere ofR^Nfor any integerN2.

• |E|denotes the Lebesgue measure of any measurable setE⊂R^N.

• −E=_|E¹_{| E}denotes the average-value over a measurable setE⊂R^N.

• For any bounded open setΩofR^N,C¹(Ω)¯ denotes the space of the restrictions toΩ¯ of the functions inC_c¹(R^N).

Note thatC¹(Ω)¯ is not generally a Banach space ifΩis not regular. But this property will not be used.

• M(X)denotes the set of the Radon measures on a locally compact setX.

2. Γ-convergence inL^∞

LetΩ be a bounded open set of R^N. Letfn, gn:Ω×R^N→ [0,∞),n∈N, be two sequences of nonnegative functions satisfying:

f_n(·, ξ ), g_n(·, ξ )are measurable for anyξ∈R^N and f_n(·,0)=g_n(·,0)=0, (2.1)

f_n(x,·), g_n(x,·)are convex for a.e.x∈Ω, (2.2)

f_n(x, ξ )g_n(x, ξ ) a.e.x∈Ω, ∀ξ∈R^N, ∀n∈N, (2.3) there existK2 and a nonnegative bounded sequenceb_ninL¹(Ω)such that

g_n(x,2ξ )Kg_n(x, ξ )+b_n a.e.x∈Ω,∀ξ∈R^N, ∀n∈N. (2.4) An easy consequence of (2.4) is the following estimate:

Proposition 2.1.There exists a constantρ1such that g_n(x, tξ )Kt^ρ

g_n(x, ξ )+b_n

a.e.x∈Ω, ∀t1,∀ξ∈R^N,∀n∈N. (2.5)

Remark 2.2. Conversely to Proposition2.1, estimate (2.5) implies (2.4) replacing K by 2^ρK. Also note that by convexity we have

g_n(x, tξ )tg_n(x, ξ ) a.e.x∈Ω,∀t∈ [0,1], ∀ξ ∈R^N, ∀n∈N. (2.6) On the other hand, taking into account the convexity ofg_nand (2.4), the following inequality holds

g_n(x, ξ+η)K 2

g_n(x, ξ )+g_n(x, η)

+b_n a.e.x∈Ω,∀ξ, η∈R^N, ∀n∈N. (2.7) Remark 2.3.Despite of the convexity and the inequalityf_ng_n, the sequencef_ndoes not satisfy in general a bound of type (2.4). Indeed, consider the following example:

Letθ:R→ [0,∞)be defined by θ (t):=

t² ift1, (t−√

k!)(√

(k+1)! +√

k!)+k! ift∈ [√ k!,√

(k+1)! ], k∈N. (2.8)

The functionθ is convex, θ (√

k!)=k!for any k∈N, andθ (t)t⁴+1 for anyt ∈R. Now define the function f_n:R^N×R^N→ [0,∞)by

f_n(x, ξ ):=a_n(x)

θ (ξ₁)+ |ξ₂|⁴+ · · · + |ξ_N|⁴

for(x, ξ )∈R^N×R^N, (2.9)

wherean:R^N→(0,∞)is a positive function. Therefore, we have f_n(x, ξ )g_n(x, ξ ):=a_n(x)

|ξ|⁴+1

for any(x, ξ )∈R^N×R^N, hence conditions (2.1)–(2.4) are clearly satisfied. However, we have forξn:=(√

n!,0, . . . ,0), f_n(2ξn)

f_n(ξ_n) =θ (2√ n!) θ (√

n!) ≈

n→∞(√

2−1)√ n,

which shows thatf_ncannot satisfy a bound of type (2.4).

LetΩbe a bounded open set ofR^N, with a Lipschitz boundary. Consider the sequence of functionalsF_ndefined by

F_n(u):=

Ωf_n(x,∇u) dx ifu∈W^1,1(Ω)∩L^∞(Ω),

∞ ifu∈L^∞(Ω)\W^1,1(Ω), G_n(u):=

Ωgn(x,∇u) dx ifu∈W^1,1(Ω)∩L^∞(Ω),

∞ ifu∈L^∞(Ω)\W^1,1(Ω). (2.10)

We make the following assumption: for anyx∈ ¯Ω, there existN+1 functionswⁱ∈C¹(Ω), 0¯ iN, such that 0 belongs to the interior of the convex envelop of

∇w⁰(x), . . . ,∇w^N(x)

, (2.11)

andN+1 sequencesw_nⁱ ∈W^1,1(Ω)∩L^∞(Ω)such that wⁱ_n→wⁱ strongly inL^∞(Ω) and G_n

w_nⁱ

is bounded. (2.12)

We have the following representation result:

Theorem 2.4. Assume that(2.1), (2.2), (2.3), (2.4), and (2.11), (2.12)hold. Then, there exist a subsequence ofn, still denoted byn, a Radon measureμ onΩ¯, and two functionsf, g: ¯Ω ×R^N→ [0,∞)satisfying the following properties:

f (·, ξ ), g(·, ξ )areμ-measurable for anyξ∈R^N and f (·,0)=g(·,0)=0, (2.13)

f (x,·), g(x,·)are convex forμ-a.e.x∈ ¯Ω, (2.14)

f (x, ξ )g(x, ξ ) μ-a.e.x∈ ¯Ω,∀ξ∈R^N, ∀n∈N, (2.15) g(x,2ξ )Kg(x, ξ )+b μ-a.e.x∈ ¯Ω,∀ξ∈R^N, ∀n∈N, (2.16) whereKis the constant in(2.4)andbis given by the convergence

b_n bμ weakly-∗inM(Ω),¯ (2.17)

such that the sequencesF_n, G_ndefined by(2.10)Γ-converge inC¹(Ω)¯ (see Definition1.2)for the strong topology of L^∞(Ω)to the functionalsF, Ggiven by

F (u):=

Ω¯

f (x,∇u) dμ, G(u):=

Ω¯

g(x,∇u) dμ, foru∈C¹(Ω).¯ (2.18)

Moreover, for any open setω⊂Ω, the sequence of functionalsF_n^ω, G^ω_n defined by F_n^ω(u):=

ωfn(x,∇u) dx ifu∈W₀^1,1(ω)∩L^∞(ω),

∞ ifu∈L^∞(ω)\W₀^1,1(ω), G^ω_n(u):=

ωg_n(x,∇u) dx ifu∈W₀^1,1(ω)∩L^∞(ω),

∞ ifu∈L^∞(ω)\W₀^1,1(ω). (2.19)

Γ-converge inC₀¹(ω)to the functionalsF^ω, G^ωgiven by F^ω(u):=

ω

f (x,∇u) dμ, G^ω(u):=

ω

g(x,∇u) dμ, foru∈C₀¹(ω). (2.20)

Remark 2.5.In the proof of Theorem2.4below we will also prove that for anyu∈C¹(Ω)¯ and any recovery se- quenceunforFnof limitu, the weak convergence of the energy density holds

f_n(·,∇u_n) f (·,∇u)μ weakly-∗inM(Ω).¯ (2.21)

Remark 2.6.Carbone and Sbordone[17]obtained a representation formula for theΓ-convergence inC⁰(Ω)of a sequence of convex functionalsF_nthe density of which satisfies

f_n(x, ξ )a_n(x)

|ξ|^p+1

a.e.x∈Ω, ∀ξ∈R^N,∀n∈N, (2.22)

wherep >1 anda_nis a bounded sequence inL¹(Ω). The condition (2.4) is sharper than (2.22). Indeed, the function g_n(x, ξ ):=a_n(x)(|ξ|^p+1)clearly satisfies the inequality (2.4). Moreover, theL¹-boundedness ofa_nin[17]is here replaced by the weaker condition (2.12). Indeed, it is easy to construct a sequencea_nwhich is not bounded inL¹(Ω) such that the extra condition (2.12) holds and for which the representation Theorem2.4applies. Think for example of the sequencef_n(x, ξ ):=(1+β_n1_B(0,n−1))|ξ|^p, whereβ_nn^−N→ ∞.

Remark 2.7.Assumptions (2.11), (2.12) are needed to ensure that the domainDF of theΓ-limitF of the sequenceFn

contains the set of regular functionsC¹(Ω). More precisely, at each point¯ x∈ ¯Ω, the gradients of(N+1)functions inC¹(Ω)¯ ∩DF have to span a sufficiently large convex set in order to derive any regular function as anL^∞-limit of a sequence of bounded energyGnin the neighborhood ofx. This is given by the barycenter condition (2.11) combined with the convergence condition (2.12) which are the key ingredients of Lemma2.10below.

Lemma 2.8.Letu_n∈W^1,1(Ω)∩L^∞(Ω)which converges strongly touinL^∞(Ω). Then, there exists a sequence

˜

u_n∈W^1,1(Ω)∩L^∞(Ω)which strongly converges touinL^∞(Ω)such that

˜

un=0 a.e. in{u=0} and fn(·,∇ ˜un)fn(·,∇un) a.e. inΩ. (2.23) Proof. Setε_n:= u_n−uL^∞(Ω). Then, the sequenceu˜_ndefined by

˜ un:=

un+εn ifun<−εn, 0 if −εnunεn, un−εn ifun> εn, clearly satisfies (2.23). 2

We have the following result:

Proposition 2.9.Assume that conditions(2.1)–(2.4), and(2.11), (2.12)hold. Then, for anyu∈C¹(Ω)¯ there exists a sequenceu_n∈W^1,1(Ω)∩L^∞(Ω)such that

u_n→ustrongly inL^∞(Ω) and G_n(u_n)is bounded. (2.24)

Proof. We need the following lemma which is a simple extension of Proposition 4.4 in[8] to dimension N 2, extendingg_n(x,·)by 0 forx∈R^N\Ω. So, we omit its proof.

Lemma 2.10.For anyx₀∈ ¯Ω, there exist three constantsε, δ, C >0such that for anyuinC¹(B(x¯ ₀, δ)∩ ¯Ω), with

∇uL^∞(B(x0,δ)∩Ω)ε, there exists a sequenceu_nsatisfying

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

un∈W^1,1

B(x0, δ)∩Ω

∩L^∞

B(x0, δ)∩Ω , un→ustrongly inL^∞

B(x0, δ)∩Ω , sup

n0

B(x₀,δ)∩Ω

gn(x,∇un) dxC.

(2.25)

Due to the compactness ofΩ¯, Lemma2.10implies the existence ofkballsB(z_i, δ_i), fori=1, . . . , k, coveringΩ¯, and constantsε, C >0 such that (2.25) holds inB(z_i, δ_i)for anyuinC¹(B(z¯ _i, δ_i)∩ ¯Ω), with∇uL^∞(B(zi,δi)∩Ω)ε, and any i=1, . . . , k. Consider a partition of the unity ϕⁱ, 1iN, such that ϕⁱ ∈C_c¹(B(z_i, δ_i)), 0ϕⁱ 1, _k

i=1ϕⁱ=1 inΩ¯. Then, there existksequencesuⁱ_ninW^1,1(B(z_i, δ_i)∩Ω)such that

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩ uⁱ_{n n}

−−−−→→∞ εϕⁱu

∇(ϕⁱu)L^∞(Ω)+1 strongly inL^∞

B(z_i, δ_i)∩Ω ,

B(zi,δi)∩Ω

g_n

·,∇uⁱ_n

dxis bounded.

(2.26)

By Lemma2.8with the open setB(z_i, δ_i)∩Ω, we can also assume thatuⁱ_n=0 a.e. in{ϕⁱ=0}. Therefore, extending uⁱ_nby 0 inΩ\B(z_i, δ_i), the sequenceu_ndefined by

un:=

k i=1

ε⁻¹∇ ϕⁱu

L^∞(Ω)+1 uⁱ_n

strongly converges touinL^∞(Ω). Moreover, by the convexity ofgnand (2.5) combined with estimate (2.26) we get that

G_n(u_n)1 k

k i=1

Ω

g_n

x, kε⁻¹∇ ϕⁱu

L∞(Ω)+1 uⁱ_n

dxc. 2

Consider for anyϕ∈C¹(Ω), the sequence of functionals¯ F_n^ϕ, G^ϕ_n defined by F_n^ϕ(v):=

Ωϕfn(x,∇v) dx ifv∈W^1,1(Ω)∩L^∞(Ω),

∞ ifv∈L^∞(Ω)\W^1,1(Ω), G^ϕ_n(v):=

Ωϕg_n(x,∇v) dx ifv∈W^1,1(Ω)∩L^∞(Ω),

∞ ifv∈L^∞(Ω)\W^1,1(Ω). (2.27)

These sequences allow us to derive local properties for theΓ-convergence of the sequencesF_n, G_n.

Proposition 2.11.Assume that conditions(2.1)–(2.4), and(2.11), (2.12)hold. Then, there exist a constantM >0and a Radon measureμonΩ¯, such that for anyϕ∈C¹(Ω)¯ withϕ0, and for anyu∈C¹(Ω), there exists a sequence¯ u_n inW^1,1(Ω)strongly converging touinL^∞(Ω)satisfying

lim sup

n→∞ G^ϕ_n(u_n)M

∇u^ρ_L∞(suppϕ)N+ ∇uL^∞(suppϕ)^N

¯ Ω

ϕ dμ. (2.28)

Proof. Define the linear functions w⁰(x):= − 1

2N N i=1

x_i and wⁱ(x):=x_i+w⁰(x), for 1iN.

By Proposition2.9there exist sequenceswⁱ_ninW^1,1strongly converging towⁱ inL^∞(Ω), withFn(wⁱ_n)bounded, for i=0, . . . , N. Define the Radon measureμonΩ¯ by

μ:=ν+ N i=0

μⁱ with

b_n ν,

g_n(·,∇wⁱ_n) μⁱ weakly-∗inM(Ω),¯ (2.29)

where theN+2 weak-∗convergences hold true up to a subsequence ofnstill denoted byn. Letu∈C¹(Ω). We can¯ assume that∇uis non-zero in suppϕ, otherwise the sequenceun:=udoes the job. Define the sequencesznandun

by

zn:=

w¹_n−w_n⁰, . . . , w^N_n −w_n⁰

∈W^1,1(Ω)^N∩L^∞(Ω)^N, u_n:=u(z_n)+4N γ

w_n⁰−w⁰(z_n)

, withγ:= ∇u_L∞(suppϕ)^N. (2.30)

Sincezn converges strongly to the identity function inL^∞(Ω)^N, the sequenceunclearly converges strongly touin L^∞(Ω). On the other hand, we have

∇u_n=4N γ _N

i=1

1 4N

2+∂_iu(z_n) γ

∇wⁱ_n+ 1 4N

2N−

N i=1

∂_iu(z_n) γ

∇w⁰_n

.

Since the term in brackets is a convex combination for large enough n(due to the strong convergence of ∂_iu(z_n) to∂_iu), the convexity ofg_ntogether with estimates (2.5) and (2.6) yields

gn(·,∇un)

K(4N γ )^ρ+4N γ bn+gn

·,∇w⁰_n +

N i=1

gn

·,∇w_nⁱ .

This combined with the definition (2.29) of the measureμimplies the desired estimate (2.28). 2

Proposition 2.12.Assume that conditions(2.1)–(2.4), and(2.11), (2.12)hold. Consideru∈C¹(Ω)¯ and a recovery sequenceu_nforF_natustrongly converging inL^∞(Ω). Then, for any sequencev_nstrongly converging touinL^∞(Ω) withF_n(v_n)bounded, and for anyϕ∈C¹(Ω)¯ withϕ0, we have

lim inf

n→∞

Ω

ϕ

fn(x,∇vn)−fn(x,∇un)

dx0. (2.31)

Remark 2.13.Proposition2.12implies some local property of theΓ-convergence ofF_n, where the strong topology ofL^∞(Ω)plays an essential role (contrary to Proposition2.11which only gives an energy bound).

Proof of Proposition 2.12. Let u∈C¹(Ω). Consider a recovery sequence¯ un for Fn strongly converging to u inL^∞(Ω). Clearly it is enough to prove the result for anyϕ∈C¹(Ω)¯ with 0ϕ1/2. By Proposition2.9there exist two sequencesϕ˜_n,ψ˜_nstrongly converging respectively toϕ,−ϕinL^∞(Ω)withF_n(ϕ˜_n), F_n(ψ˜_n)bounded. Then, the truncations ϕ_n:=(0∨ ˜ϕ_n)∧1/2,ψ_n:=(−1/2∨ ˜ψ_n)∧0 strongly converge respectively toϕ,−ϕ inL^∞(Ω), and satisfy

0ϕn,−ψn1/2 and gn(·, ϕn)gn(·,ϕ˜n), gn(·, ψn)gn(·,ψ˜n)a.e. inΩ, so that the energiesG_n(ϕ_n), G_n(ψ_n)are bounded.

Consider a sequence v_n∈W^1,1(Ω) strongly converging tou inL^∞(Ω) withF_n(v_n)bounded and a sequence z_n∈W^1,1(Ω)strongly converging touinL^∞(Ω)withG_n(z_n)bounded, and define forε∈(0,1/2)the sequence

w_n:=(1−ε)u_n+ϕ_n(v_n−u_n)⁺+ψ_n(v_n−u_n)⁻+εz_n. In the set{u_nv_n}we have

∇w_n=(1−ε−ϕ_n)∇u_n+ϕ_n∇v_n+ε

∇z_n+ε⁻¹(v_n−u_n)∇ϕ_n

=(1−ε−ϕ_n)∇u_n+ϕ_n∇v_n+ε

1−(v_n−u_n)

∇z_n+(v_n−u_n)

∇z_n+ε⁻¹∇ϕ_n .

Note that the last term is a linear combination fornlarge enough, since we work in the set{u_nv_n}andv_n−u_n converges strongly inL^∞(Ω). Then, by the convexity off_nand (2.3), (2.7) we get that

{unvn}

f_n(x,∇w_n) dx

{unvn}

(1−ε−ϕ_n)f_n(x,∇u_n)+ϕ_nf_n(x,∇v_n) dx

+ε

{u_nv_n}

1−(vn−un)

gn(x,∇zn)+(vn−un)gn

x,∇zn+ε⁻¹∇ϕn

dx.

From the estimates (2.5), (2.6), (2.7) satisfied byg_n, the boundedness ofF_n(u_n),F_n(v_n),G_n(z_n),G_n(ϕ_n), and the strong convergence ofϕ_ntoϕandv_n−u_nto 0 inL^∞(Ω), we deduce that

{unvn}

f_n(x,∇w_n) dx

{unvn}

(1−ε−ϕ)f_n(x,∇u_n)+ϕf_n(x,∇v_n) dx+ε

{unvn}

g_n(x,∇z_n) dx+o(1), whereo(1)tends to 0 asn→ ∞for a fixedε∈(0,1/2). Similarly for the set{v_n< u_n}with the sequenceψ_n, we obtain that

{v_n<u_n}

f_n(x,∇w_n) dx

{v_n<u_n}

(1−ε−ϕ)f_n(x,∇u_n)+ϕf_n(x,∇v_n) dx+ε

{v_n<u_n}

g_n(x,∇z_n) dx+o(1).

Using that w_n converges strongly to uinL^∞(Ω)and that u_n is a recovery sequence forF_n, and adding the two previous inequalities, it follows that

Ω

fn(x,∇un) dx

Ω

fn(x,∇wn) dx+o(1)

Ω

fn(x,∇un) dx+

Ω

ϕ

fn(x,∇vn)−fn(x,∇un) dx +ε

Ω

gn(x,∇zn)−fn(x,∇un)

dx+o(1), which implies

lim inf

n→∞

Ω

ϕ

f_n(x,∇v_n)−f_n(x,∇u_n)

dx−εlim sup

n→∞

Ω

g_n(x,∇z_n)−f_n(x,∇u_n) dx.

Finally, the arbitrariness ofεyields (2.31). 2

Proposition 2.14.Assume that(2.1)–(2.4), and(2.11), (2.12)hold. Then, there exists a constantC >0such that for anyu, v∈C¹(Ω)¯ and any recovery sequences u_n, v_n forF_nrespectively atu, v, converging respectively tou, vin L^∞(Ω), and for anyϕ∈C_c¹(Ω)¯ withϕ0, the following estimate holds

lim sup

n→∞

Ω

ϕ

fn(x,∇un)−fn(x,∇vn) dx

C∇u− ∇vL^∞(suppϕ)

∇u^ρ_L∞(suppϕ)+ ∇v^ρ_L∞(suppϕ)+1

¯ Ω

ϕ dμ. (2.32)

Proof. Let u, v, ϕ∈C¹(Ω), and set¯ γ := ∇u− ∇vL^∞(suppϕ). Consider two recovery sequences un, vn for Fn

respectively at u, v, converging respectively to u, v inL^∞(Ω). First, note that by virtue of Proposition2.11 and Proposition2.12(applied to the recovery sequencesu_n, v_n) combined with the inequalityF_n^ϕG^ϕ_n, estimate (2.32) holds whenγ1. From now on, we assume thatγ <1.

Takeε∈(0,1−γ ). By Proposition2.11there exists a sequenceζ_n∈W^1,1(Ω)strongly converging toζ :=v+ (γ+ε)⁻¹(u−v)inL^∞(Ω), and satisfying the bound (2.28) with the pair(ζn, ζ ). The sequenceu˜n:=(1−γ−ε)vn+ (γ +ε)ζ_nconverges strongly touinL^∞(Ω). Then, sinceu_nis a recovery sequence forF_n, by Proposition2.12and by the convexity off_nwe have

F_n^ϕ(u_n)F_n^ϕ(u˜_n)+o(1)(1−γ−ε)F_n^ϕ(v_n)+(γ +ε)G^ϕ_n(ζ_n)+o(1) F_n^ϕ(v_n)+M(γ+ε)

∇ζ^ρ_L∞(suppϕ)N+ ∇ζL^∞(suppϕ)^N Ω¯

ϕ dμ+o(1)

F_n^ϕ(v_n)+2^ρM(γ +ε)

∇v^ρ_L∞(suppϕ)N+1

¯ Ω

ϕ dμ+o(1).

Changing the roles ofu_nandv_nwe also get that F_n^ϕ(v_n)F_n^ϕ(u_n)+2^ρM(γ +ε)

∇u^ρ_L∞(suppϕ)N+1

¯ Ω

ϕ dμ+o(1).

Therefore, from the two previous inequalities we deduce that lim sup

n→∞ F_n^ϕ(u_n)−F_n^ϕ(v_n)2^ρM(γ+ε)

∇u^ρ_L∞(suppϕ)N+ ∇v^ρ_L∞(suppϕ)N+2

¯ Ω

ϕ dμ.

Finally, the arbitrariness ofε >0 implies the desired estimate (2.32). 2

Proof of Theorem2.4. First, note that it is enough to prove the results for the sequencef_n, since the properties off_n are clearly satisfied by g_n. Thanks to Proposition2.9, Proposition2.11, and using a diagonal extraction there exists a subsequence ofnstill denoted byn, such that for any linear functionw^ξ :x→ξ ·x, withξ ∈Q^N, there exists a recovery sequencew^ξ_nforF_natw^ξstrongly converging inL^∞(Ω)tow^ξ, and a functionh^ξ∈L¹_μ(Ω)¯ such that

f_n

·,∇w^ξ_n

h^ξμ weakly-∗inM(Ω),¯ (2.33)

where by estimates (2.28) and (2.32) the functionh^ξsatisfies 0h^ξM

|ξ|^ρ+ |ξ|

μ-a.e. inΩ,¯ ∀ξ∈Q^N, h^ξ−h^ηC|ξ−η|

|ξ|^ρ+ |η|^ρ+1

μ-a.e. inΩ,¯ ∀ξ, η∈Q^N. (2.34)

By the Lipschitz estimate of (2.34) there exists a unique Caratheodory functionf : ¯Ω×R^N→ [0,∞)defined by

f (x, ξ ):=h^ξ(x) μ-a.e.x∈ ¯Ω,∀ξ∈Q^N. (2.35)

First step:Γ-convergence ofF_ninC¹(Ω).¯

Let u∈C¹(Ω). There exists a subsequence¯ n of n such thatF_n Γ-converges at u. Consider a recovery se- quence u_n forF_n, which strongly converges tou inL^∞(Ω). Up to extract a new subsequence, thanks to Propo- sition 2.11and Proposition2.12combined with F_n^ϕG^ϕ_n, we can also assume that there existsh^u∈L¹_μ(Ω)¯ such that

f_n(·,∇u_n) h^uμ weakly-∗inM(Ω).¯ (2.36)

Applying (2.32) (after a localization) with the recovery sequencesu_n, w_n^ξ, forξ∈Q^N, we obtain that h^u−f (·, ξ )C|∇u−ξ|

|∇u|^ρ+ |∇u| + |ξ|^ρ+ |ξ| +1

μ-a.e. inΩ,¯ ∀ξ∈Q^N,

which by continuity off (x,·)implies thath^u=f (·,∇u)a.e. inΩ¯. Hence, by convergence (2.36) and a uniqueness argument the whole sequenceF_n Γ-converges atutoF (u)defined by (2.18). This also implies convergence (2.21) for any recovery sequenceu_nforF_n.

Second step: Properties of the densityf.

Let us prove the convexity off (x,·). The proofs of the other properties are similar. Letξ, η∈R^N, and setλ:=

t ξ+(1−t )ηfor t ∈ [0,1]. Consider recovery sequencesw_n^ξ,w_n^η,w^λ_n, for F_n of limitsw^ξ, w^η, w^λ respectively.

Applying inequality (2.31) withu_n:=w^λ_nandv_n:=t w^ξ_n+(1−t )w^η_nand using the convergence (2.21) of the energy density, we get that for anyϕ∈C¹(Ω)¯ withϕ0,

¯ Ω

ϕf

x, tξ+(1−t )η

dμ= lim

n→∞

Ω

ϕf_n

x,∇w_n^λ dx

lim inf

n→∞

Ω

ϕf_n

x, t∇w_n^ξ+(1−t )∇w^η_n dx t

¯ Ω

ϕf (x, ξ ) dμ+(1−t )

¯ Ω

ϕf (x, η) dμ,

which implies the convexity off (x,·)due to the arbitrariness ofϕ.

Third step:Γ-convergence ofF_n^ω for any open setω⊂Ω.

Let us prove theΓ-liminf and theΓ-limsup properties for the sequenceF_n^ω (see Definition1.2). Letu∈C₀¹(ω).

Letu_n be a sequence ofW₀^1,1(ω)which strongly converges touinL^∞(ω). Extendinguandu_n by 0 inΩ\ω, the Γ-liminf property forF_nimplies that

F (u)lim inf

n→∞ F_n(u_n)=lim inf

n→∞ F_n^ω(u_n).