Γ-convergence of nonconvex unbounded integrals in Cheeger-Sobolev spaces

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Γ-convergence of nonconvex unbounded integrals in Cheeger-Sobolev spaces

Omar Anza Hafsa, Jean-Philippe Mandallena

To cite this version:

Omar Anza Hafsa, Jean-Philippe Mandallena. Γ-convergence of nonconvex unbounded integrals in Cheeger-Sobolev spaces. 2020. �hal-02295632v2�

ΓΓΓ-CONVERGENCE OF NONCONVEX UNBOUNDED INTEGRALS IN CHEEGER-SOBOLEV SPACES

OMAR ANZA HAFSA AND JEAN-PHILIPPE MANDALLENA

Abstract. We study Γ-convergence of nonconvex integrals of the calculus of variations in the setting of Cheeger-Sobolev spaces when the integrands have not polynomial growth and can take infinite values. Applications to relaxation and homogenization are also developed.

Contents

1. Introduction 2

2. The Γ-convergence result 4

3. Auxiliary results 7

3.1. Cheeger-Sobolev spaces 7

3.2. Ru-usc integrands 12

3.3. Integral representation of the Vitali envelope of a set function 15

4. Proofs 15

4.1. Proof of the lower bound 15

4.2. Proof of the upper bound 21

4.3. Proof of the Γ-convergence result 29

5. Applications 30

5.1. Relaxation 30

5.2. Homogenization 31

References 39

Key words and phrases. Γ-convergence, Nonconvex unbounded integral, Ru-usc, General growth condi- tions, Metric measure space, Cheeger-Sobolev space.

1

OMAR ANZA HAFSA AND JEAN-PHILIPPE MANDALLENA

1. Introduction

Let pX, d, µq be a metric measure space, where pX, dq is separable and complete and µis a doubling positive Radon measure onXwhich satisfies the annular decay property, supporting a weak p1, pq-Poincar´e inequality with p ą1. Let m, N ě 1 be two integers, let O Ă X be a bounded open set such that µpOzOq “0 and let pΩ,F,Pq be a probability space. In this paper we consider a family of stochastic integralsE_t :H_µ^1,ppO;R^mq ˆΩ!r0,8s defined by

E_tpu, ωq:“ ż

O

L_t`

x,∇_µupxq, ω˘

dµpxq, (1.1)

where L_t:OˆMˆΩ!r0,8s is a Borel measurable stochastic integrand¹ depending on a parameter t ą0, not necessarily convex with respect to ξ PM, where M denotes the space of real mˆN matrices, and possibly taking infinite values. The space H_µ^1,ppO;R^mqdenotes the class ofp-Cheeger-Sobolev functions from Ω to R^m and ∇_µu is the µ-gradient of u.

The object of the present paper is to deal with the problem of computing the almost sure Γ-convergence (see Definitions 2.1) of the stochastic familytE_tutą0, ast!8, to a stochastic integral E_lim:H_µ^1,ppO;R^mq ˆΩ!r0,8s of the type

E_limpu, ωq “ ż

O

L_lim`

x,∇_µupxq, ω˘

dµpxq (1.2)

with L_lim : OˆMˆΩ! r0,8s not depending on the parameter t. When L_lim is indepen- dent of the variable x, the procedure of passing from (1.1) to (1.2) is referred as stochastic homogenization. If furthermore Llim is independent of the variableω then Elim is said to be deterministic, otherwise Elim is said to be stochastic. When tLtutą0 is deterministic, i.e. Lt

is independent of the variable ω for all t ą0, the procedure of passing from (1.1) to (1.2) is referred as deterministic homogenization.

This Γ-convergence problem was already studied in [AHM17] in the case where L_t has p- growth. Here we treat the case where Lt has not necessarily p-growth and can take infinite values (see Sect. 2 for more details).

For related works in the Euclidean case, i.e. when pX, d, µq “ pR^N,| ¨ ´ ¨ |,LNq where LN is the Lebesgue measure onR^N, we refer the reader to [Mar78, Bra85, DMM86, M¨ul87, JKO94, MM94, BG95, BD98, AM02, AM04, AHM11, AHLM11, AHMZ15, DG16, AHCM17] and the references therein.

One motivation for developing Γ-convergence, and more generally calculus of variations, in the setting of metric measure spaces comes from applications to hyperelasticity. In fact, the interest of considering a general measure is that its support can be interpretated as a hyperelastic structure together with its singularities like for example thin dimensions, corners, junctions, etc. Such mechanical “singular” objects naturally lead to develop calculus of variations in the setting of metric measure spaces. Indeed, for example, a low multi- dimensional structures can be described by a finite number of smooth compact manifoldsSi 1Throughout the paper, by a Borel measurable stochastic integrandL:OˆMˆΩ!r0,8swe mean that LispBpXq bBpMq bF,BpRqq-measurable, whereBpXq,BpMqandBpRqdenote the Borelσ-algebra on X,MandRrespectively.

of dimensionk_i on which a superficial measureµ_i “H^ki|Si is attached. Such a situation leads to deal with the finite union of manifolds S_i, i.e. X “ YiS_i, together with the finite sum of measuresµ_i, i.e. µ“ř

iµ_i, whose mathematical framework is that of metric measure spaces (for more examples, we refer the reader to [BBS97, Zhi02, CJLP02] and [CPS07, Chapter 2,

§10] and the references therein).

Another motivation is the development of the calculus of variations on “singular” spaces, which are of interest for geometers and physicists, like Carnot groups, glued spaces, Laakso spaces, Bourdon-Pajot spaces, Gromov-Hausdorff limit spaces, spaces statisfying generalized Ricci bounds (see [KM16] for more details). Indeed, all these spaces are examples of doubling metric measure spaces satisfying a Poincar´e inequality on which the theory of Γ-convergence on Cheeger-Sobolev spaces could be applied.

The plan of the paper is as follows. In Sect. 2 we state the main result of the paper, see Theorem 2.5 (and also Proposition 2.8 whose proof is given at the end of Sect. 2). Theorem 2.5 is a Γ-convergence result of tE_tutą0 as t ! 8 to E_lim in the setting of metric measure spaces and in a unbounded framework. Classically, its proof is a consequence of Proposition 2.6 (the lower bound) and Proposition 2.7 (the upper bound). Sect. 3 is devoted to several auxiliary definitions and results needed for understanding and proving our Γ-convergence result: in §3.1 we provide materials about Cheeger-Sobolev spaces; in §3.2 we recall the concept of (family of) ru-usc² integrand(s) and its main properties that will be used in the proof of Propositions 2.6, 2.7 and 2.8; the proof of Proposition 2.7 also needs the use of the Viatli envelope of a set function which is recalled in §3.3. Sect. 4 is devoted to the proofs of Propositions 2.6 and 2.7 and Theorem 2.5. Finally, applications to relaxation and homogenization are developed in Sect. 5.

Notation. The open and closed balls centered atxP X with radius ρą0 are denoted by:

B_ρpxq:“

!

yP X :dpx, yq ăρ )

; Bρpxq:“

!

yP X :dpx, yq ď ρ )

. ForxP X and ρą0 we set

BB_ρpxq:“B_ρpxqzB_ρpxq “

!

yPX :dpx, yq “ ρ )

.

For A ĂX, the diameter of A (resp. the distance from a point x P X to the subset A) is defined by diampAq:“sup_x,yPAdpx, yq.

The symbol ´ş

stands for the mean-value integral

´ ż

B

f dµ“ 1 µpBq

ż

B

f dµ.

ForFĂM, where M denotes the space of realmˆN matrices, the interior and the closure of F are respectively denoted by intpFq and F.

2The abbreviation ru-usc means radially uniformly upper semicontinuous.

OMAR ANZA HAFSA AND JEAN-PHILIPPE MANDALLENA

2. The Γ-convergence result

We begin by recalling the definition of the almost sure Γ-convergence. (For more details on the theory of Γ-convergence we refer to [DM93].)

Definition 2.1. We say that tE_tutą0 almost surely ΓpL^p_µq-converges as t! 8 to the func- tional E_lim : H_µ^1,ppO;R^mq ˆΩ! r0,8s if there exists Ω¹ P F with PpΩ¹q “ 1 such that for every ω PΩ¹, one has:

Γ-lim: for every uP H_µ^1,ppO;R^mq, ΓpL^p_µq- lim_t!8E_tpu, ωq ě E_limpu, ωqwith ΓpL^p_µq- lim

t!8

E_tpu, ωq:“inf

"

lim

t!8

E_tpu_t, ωq:u_t!u inL^p_µpO;R^mq

* ,

or equivalently, for everyuPH_µ^1,ppO;R^mqand everytu_tutą0 ĂH_µ^1,ppO;R^mqsuch that u_t!u inL^p_µpO;R^mq,

lim

t!8E_tpu_ε, ωq ě E_limpu, ωq;

Γ-lim: for every uP H_µ^1,ppO;R^mq, ΓpL^p_µq- lim_t!8E_tpu, ωq ď E_limpu, ωqwith ΓpL^p_µq- lim

t!8E_tpu, ωq:“inf

!

tlim!8E_tpu_t, ωq:u_t!u inL^p_µpO;R^mq )

,

or equivalently, for every u P H_µ^1,ppO;R^mq there exists tututą0 Ă H_µ^1,ppO;R^mq such that u_t!u inL^p_µpO;R^mq and

tlim!8E_tpu_t, ωq ďE_limpu, ωq.

Referring to the next section for any unfamiliar notation or definition, in what follows we state the main results of the paper. Let m, N ě 1 be two integers and let M be the space of real mˆN matrices. Let G:M!r0,8s be a Borel measurable integrand satisfying the following conditions:

(C₁) there exists γ ą0 such that for everyξ, ζ PMand every τ Ps0,1r, Gpτ ξ` p1´τqζq ďγp1`Gpξq `Gpζqq;

(C₂) 0P intpGq, whereGdenotes the effective domain ofG, i.e. G:“ tξ PM:Gpξq ă 8u.

Remark 2.2. If (C₁) is satisfied then G is convex. If moreover (C₂) holds, then there exists rą0 such that sup_|ξ|ďrGpξq ă 8,see [AHM12b, Lemma 4.1].

LetQ_µG:OˆM!r0,8s be defined by QµGpx, ξq:“lim

ρ!0inf

#

´ ż

Bρpxq

Gpξ`∇_µwpyqqdµpyq:wPH_µ,0^1,ppB_ρpxq;R^mq +

,

where the space H_µ,0^1,ppB_ρpxq;R^mqis defined as the closure of Lip₀pB_ρpxq;R^mq:“

!

uPLippO;R^mq:u“0 on OzB_ρpxq )

with respect to the H_µ^1,p-norm, where LippO;R^mq :“ rLippOqs^m with LippOq denoting the algebra of Lipschitz functions from O to R. (The integrand Q_µG is called the H_µ^1,p- quasiconvexification of G. For more details on the notion of H_µ^1,p-quasiconvexity, we refer to [AHM20a].) Denote the the effective domain of Q_µGpx,¨q byQ_µGx. We further suppose that:

(C₃) for every xPO, τQ_µGx ĂintpQ_µGxqfor all τ Ps0,1r;

(C4) for every u P H_µ^1,ppO;R^mq, if ş

OQµGpx,∇µupxqqdµ ă 8 and if ∇µupxq P intpQµG^xq forµ-a.a. xPO then ş

OGp∇_µupxqqdµă 8; (C₅) for every xPO,Q_µGpx,¨q is lsc³ on intpQ_µGxq.

Remark 2.3. Under (C₁) and (C₂) if G “ Q_µG, i.e. G is H_µ^1,p-quasiconvex, then (C₃) and (C₄) hold. In particular, since convexity implies H_µ^1,p-quasiconvexity (see [AHM20a]), if G is convex then (C₃) and (C₄) hold.

Let pX, d, µq be a metric measure space, where pX, dq is separable and complete and µis a doubling positive Radon measure onXwhich satisfies the annular decay property, supporting a weak p1, pq-Poincar´e inequality with

pąκ:“ lnpC_dq

lnp2q where C_dě1 is the doubling constant.

LetO ĂX be a bounded open set such that µpOzOq “0 and let pΩ,F,Pq be a probability space. Throughout the paper, we consider a family tLt : O ˆM ˆΩ ! r0,8sutą0 of Borel measurable stochastic integrands depending on a parameter t ą 0 (destined to tend to infinity) and satisfying the following conditions:

(C6) tLtutą0 is p-coercive, i.e. there exists c ą0 such that for every t ą0, every x P O, every ξP Mand every ωPΩ,

L_tpx, ξ, ωq ěc|ξ|^p;

(C₇) tL_tutą0 hasG-growth, i.e. there existα, β ą0 such that for everyxPO, every ξPM and everyω PΩ,

αGpξq ďL_tpx, ξ, ωq ď βp1`Gpξqq.

Remark 2.4. If (C₁) and (C₇) hold then the effective domain Lt,x,ω of L_tpx,¨, ωq is equal to G and so is convex and does not depend on x and ω.

The p-growth case, i.e. when Gpξq “ |ξ|^p, was already studied in [AHM17]. The object of this paper is to deal with the G-growth case. For this, in addition, we need to suppose that (C₈) for everyω PΩ,tL_tutą0 is ru-usc atω, i.e. for everyωPΩ, there existsta_tp¨, ωqu_tą0 Ă

L¹_µpO;s0,8sq with

tlim!8

ż

O

a_tpx, ωqdµpxq ă 8 (2.1)

3The abbreviation lsc means lower semicontinuous.

OMAR ANZA HAFSA AND JEAN-PHILIPPE MANDALLENA

and

ρlim!0 lim

t!8´ ż

Bρp¨q

a_tpy, ωqdµpyq “:a₈p¨, ωq PL¹_µpOq (2.2) such that

lim

τ!1^´sup

tą0

∆^a_L^t_tpτ, ωq ď0, where ∆^a_L^t

t :r0,1s ˆΩ!s ´ 8,8s is given by

∆^a_L^t

tpτ, ωq:“sup

xPO

sup

ξPLt,x,ω

L_tpx, τ ξ, ωq ´L_tpx, ξ, ωq

a_tpx, ωq `L_tpx, ξ, ωq (2.3) with Lt,x,ω denoting the effective domain of L_tpx,¨, ωq.

For each tą0 and each ρą0, letH_µ^ρLt :OˆMˆΩ!r0,8s be defined by H_µ^ρLtpx, ξ, ωq:“inf

#

´ ż

Bρpxq

Ltpy, ξ`∇µwpyq, ωqdµpyq:wPH_µ,0^1,ppBρpxq;R^mq +

. For each t ą0, let E_t : H_µ^1,ppO;R^mq ˆΩ ! r0,8s be defined by (1.1). The main result of the paper is the following Γ-convergence result.

Theorem 2.5 (Γ- lim). Assume that pąκ. If (C₁)–(C₈) hold and if

(C₉) there exists Ω¹ PF with PpΩ¹q “1 such that for every ω PΩ¹, one has limρ!0lim

t!8

H_µ^ρL_tpx, ξ, ωq ě lim

ρ!0lim

t!8H_µ^ρL_tpx, ξ, ωq for all xPO and all ξ PQ_µGx,

then tE_tutą0 almost surely ΓpL^p_µq-converges as t !8 to the functional E_lim:H_µ^1,ppO;R^mq ˆ Ω!r0,8s defined by (1.2) with Llim:OˆMˆΩ!r0,8s given by

L_limpx, ξ, ωq “ lim

τ!1^´

limρ!0lim

t!8H_µ^ρL_tpx, τ ξ, ωq Theorem 2.5 is a consequence of the following two propositions.

Proposition 2.6 (Γ- lim). Assume that p ąκ. If (C1), (C2) and (C6)–(C8) hold then, for every ω PΩ, one has

ΓpL^p_µq- lim

t!8

E_tpu, ωq ě ż

O

lim

τ!1^´

ρlim!0 lim

t!8

H_µ^ρL_tpx, τ∇_µupxq, ωqdµpxq for all uPH_µ^1,ppO;R^mq.

Proposition 2.7 (Γ- lim). Assume thatpąκ. If (C₁)–(C₈) hold then, for everyω PΩ, one has

ΓpL^p_µq- lim

t!8E_tpu, ωq ď ż

O

lim

τ!1^´

ρlim!0 lim

t!8H_µ^ρL_tpx, τ∇_µupxq, ωqdµpxq for all uPH_µ^1,p`

O;R^m˘ .

LetL₈ :OˆMˆΩ!r0,8s be defined by L₈px, ξ, ωq:“lim

ρ!0lim

t!8H_µ^ρL_tpx, ξ, ωq.

(Note that if (C₉) is satisfied thenL₈p¨,¨, ωq “lim_ρ!0lim_t!8H_µ^ρL_tp¨,¨, ωqforP-a.e. ωP Ω.) LetLp₈ :OˆMˆΩ!r0,8s be given by

Lp₈px, ξ, ωq:“ lim

τ!1^´

L₈px, τ ξ, ωq

and, for each x P O and each ω P Ω, let L₈px,¨, ωq denotes the lsc envelope of L₈px,¨, ωq.

The following proposition makes more precise the formula of the limit integrand L_lim in Theorem 2.5.

Proposition 2.8. Assume that (C₃), (C₇) and (C₈) hold.

(i) For every ωP Ω, Lp₈px, ξ, ωq “ lim

τ!1^´L₈px, τ ξ, ωq “

# lim

τ!1^´L₈px, τ ξ, ωq if ξPQ_µGx

8 otherwise.

So, in Theorem2.5 we have L_lim “Lp₈.

(ii) Suppose furthermore that for every ω P Ω and every x P O, L₈px,¨, ωq is lsc on intpQ_µGxq. Then

Lp₈px, ξ, ωq “ L₈px, ξ, ωq “

$

&

%

L₈px, ξ, ωq if ξ PintpQ_µGxq lim

τ!1^´L₈px, τ ξ, ωq if ξ P BQ_µGx

8 otherwise.

(2.4) In such a case, in Theorem 2.5, L_lim is given by (2.4).

Proof of Proposition 2.8. From (C8) and proposition 3.14, we can assert that for every ω PΩ,L₈ is ru-usc at ω. Moreover, by (C7) it is easily seen that for everyxP O and every ω P Ω, the effective domain of L₈px,¨, ωq is equal toQµG^x. So, taking (C₃) into account, Proposition 2.8 follows from Theorem 3.12.

3. Auxiliary results

3.1. Cheeger-Sobolev spaces. Let pX, d, µq be a separable and complete metric measure space. Here and subsequently, we assume thatµis doubling onX, i.e. there exists a constant C_dě1 such that

µpB_ρpxqq ď C_dµ

´ B^ρ

2pxq

¯

(3.1) forµ-a.a. xP X and allρą0, andX supports a weak p1, pq-Poincar´e inequality withpą1, i.e. there exist C_P ą0 and σ ě1 such that for µ-a.e. xPX and every ρą0,

´ ż

Bρpxq

ˇ ˇ ˇ ˇ ˇ

u´ ´ ż

Bρpxq

udµ ˇ ˇ ˇ ˇ ˇ

dµďρC_P

˜

´ ż

Bσρpxq

v^pdµ

¸¹_p

(3.2)

OMAR ANZA HAFSA AND JEAN-PHILIPPE MANDALLENA

for everyuPL^p_µpOq, everyp-weak upper gradient⁴v PL^p_µpOqforuand every open setO ĂX such that B_σρpxq ĂO.

Remark 3.1. Asµis doubling, forµ-a.e. ¯xP Xand everyrą0, we haveµpB_ρpxqq{µpB_rp¯xqq ě 4^´κpρ{rq^κ for all xPB_rp¯xqand all 0 ăρďr, where κ:“ ^lnpC_lnp2q^dq (see [Haj03, Lemma 4.7]).

We further assume that pX, d, µq satisfies the annular decay property, i.e. there exist δ ą0 and C_Aě1 such that

µpB_σrpxqzB_rpxqq ď C_A ˆ

1´ 1 σ

˙δ

µpB_σrpxqq (3.3)

for all xPX, all rą0 and all σPs1,8r.

Remark 3.2. From [Buc99, Corollary 2.2] and [CM98, Lemma 3.3] (see also [Che99, Propo- sition 6.12] and [HKST15, Proposition 11.5.3 pp. 328]), under (3.1) and (3.2), if moreover pX, dq is a length space, i.e. the distance between any two points equals infimum of lengths of curves connecting the points, then (3.3) holds.

Remark 3.3. If (3.3) holds then µ`

B_rpxqzB_rpxq˘

“ 0 for all x P X and all r ą 0, i.e.

the boundary of any ball is of zero measure. Indeed, given x P X and r ą 0, we have 1 ě ^µpB_µpB^rpxqq

rpxqq ě _µpB^µpB_σr^rpxqq_pxqq ě 1´C_Ap1´ _σ¹q^δ for all σ Ps1,8r. Hence, by letting σ ! 1, we obtain ^µpBrpxqq

µpBrpxqq “1, i.e. µpB_rpxqq “µpB_rpxqq.

Let O Ă X be a bounded open set. Denote the algebra of Lipschitz functions from O to R by LippOq. (Note that, by Hopf-Rinow’s theorem (see [BH99, Proposition 3.7, pp. 35]), the closure of O is compact, and so every Lipschitz function from O to R is bounded.) Let LippO;R^mq:“ rLippOqs^m and let ∇_µ: LippO;R^mq!L⁸_µpO;Mqbe given by

∇_µu:“

¨

˝ D_µu₁

... D_µu_m

˛

‚with u“ pu₁,¨ ¨ ¨, u_mq,

where D_µ : LippOq ! L⁸_µpO;R^Nq is the differential of Cheeger (see [Che99, Theorem 4.38]

and [Kei04, Definition 2.1.1 and Theorem 2.3.1] for more details). The p-Cheeger-Sobolev space H_µ^1,ppO;R^mq is defined as the completion of LippO;R^mqwith respect to the norm

}u}_Hµ^1,p_pO;Rmq:“ }u}L^pµpO;R^mq` }∇_µu}L^pµpO;Mq. (3.4) As}∇_µu}L^pµpO;Mq ď }u}_Wµ^1,p_pO;Rmqfor alluP LippO;R^mq, the linear map∇_µfrom LippO;R^mq to L^p_µpO;Mq has a unique extension to H_µ^1,ppO;R^mq which will still be denoted by ∇_µ and will be called the µ-gradient. For more details on the various possible extensions of the

4A Borel function v :O !r0,8s is said to be an upper gradient for u:O !Rif |upcp1qq ´upcp0qq| ď ş1

0vpcpsqqdsfor all continuous rectifiable curvesc:r0,1s!O. A function vPL^p_µpOqis said to be ap-weak upper gradient foruPL^p_µpOq if there existtu_nun ĂL^p_µpOq andtv_nun ĂL^p_µpOq such that for eachně1, vn is an upper gradient for un, un ! u in L^p_µpOq and vn ! v in L^p_µpOq. For more details we refer to [HK98, Che99].

classical theory of the Sobolev spaces to the setting of metric measure spaces, we refer to [Hei07, §10-14] (see also [Che99, Sha00, GT01, Haj03]).

The following proposition brings together useful known properties for dealing with calculus of variations in the metric measure setting. (For a proof we refer to [HKST15] and [AHM20a,

§7].)

Proposition 3.4. Under (3.1), (3.2) and (3.3) the following properties hold:

(i) O satisfies the Vitali covering theorem, i.e. for every A ĂO and every family B of closed balls in O, if inftρ ą 0 : B_ρpxq P Bu “ 0 for all x P A (we say that B is a fine cover of A) then there exists a countable disjoint subfamily B¹ of B such that µpAz YBPB¹Bq “ 0; in other words, AĂ`

YBPB¹B˘

YN with µpNq “0;

(ii) the µ-gradient is closable in H_µ^1,ppO;R^mq, i.e. for every u P H_µ^1,ppO;R^mq and every open set AĂO, if upxq “ 0 for µ-a.a. xP A then ∇_µupxq “ 0 for µ-a.a. xP A;

(iii) O supports a p-Sobolev inequality, i.e. there exists C_S ą0 such that

˜ż

Bρpxq

|v|^pdµ

¸¹

p

ďρC_S

˜ż

Bρpxq

|∇_µv|^pdµ

¸¹

p

for all0ăρďρ0, withρ0 ą0, and all v PH_µ,0^1,ppBρpxq;R^mq, where, for each open set AĂO, H_µ,0^1,ppA;R^mq is the closure of Lip₀pA;R^mq with respect to H_µ^1,p-norm defined in (3.4) with

Lip₀pA;R^mq:“ uPLippO;R^mq:u“0 on OzA(

;

(iv) for every uPH_µ^1,ppO;R^mq and µ-a.e. xPO there exists u_x PH_µ^1,ppO;R^mq such that:

∇_µu_xpyq “ ∇_µupxq for µ-a.a. y PO;

limρ!0

1

ρ}u´u_x}L⁸_µpBρpxq;R^mq “0 if pąκ, where κ:“ ^lnpC_lnp2q^dq with C_d ě1 given by the inequality (3.1);

(v) for every x P O, every ρ ą 0 and every λ Ps0,1r there exists a Urysohn function ϕPLippOq for the pair pOzB_ρpxq, B_λρpxqq⁵ such that

}D_µϕ}_L8

µpO;R^Nq ď θ ρp1´λq for someθ ą0;

(vi) for µ-a.e. xP O, lim

λ!1^´

lim

ρ!0

µpB_λρpxqq

µpB_ρpxqq “ lim

λ!1^´

limρ!0

µpB_λρpxqq µpB_ρpxqq “1;

5Given a metric spacepO, dq, by a Urysohn function fromOtoRfor the pairpOzV, Kq, whereKĂV ĂO withKcompact andV open, we mean a continuous functionϕ:O!Rsuch thatϕpxq P r0,1sfor allxPO, ϕpxq “0 for allxPOzV andϕpxq “1 for allxPK.

OMAR ANZA HAFSA AND JEAN-PHILIPPE MANDALLENA

(vii) for every uPH_µ^1,ppO;R^mq and every ϕPLippOq,

∇_µpϕuq “ϕ∇_µu`D_µϕbu.

Remark 3.5. Asµis a Radon measure and O satisfies the Vitali covering theorem, for every open setAĂO and every εą0 there exists a countable familytB_ρipx_iquiPI of disjoint open balls of A with x_i P A, ρ_i Ps0, εr such that µ`

Az YiPI B_ρipx_iq˘

“ 0. By the annular decay property, see (3.3), we also have µpBBρipxiqq “0 for all iPI (see Remark 3.3).

In the framework of thep-Cheeger-Sobolev spaces withpąκ :“lnpC_dq{lnp2q, whereC_d ě1 is the doubling constant, we also have the following L⁸_µ-compactness result.

Theorem 3.6. Assume that p ą κ and µ` OzO˘

“ 0. If u P H_µ^1,ppO;R^mq and tu_nun Ă H_µ^1,ppO;R^mq are such that

nlim!8}u_n´u}L^pµpO;R^mq “0 and sup

ně1

}∇_µu_n}L^pµpO;Mqă 8, (3.5) then, up to a subsequence,

nlim!8}u_n´u}_L⁸_µpO;R^m_q “0. (3.6) Proof of Theorem 3.6. Since pX, d, µq is a complete doubling metric space, pX, d, µq is proper, i.e. every closed ball is compact (see [HKST15, Lemma 4.1.14]), and so pO, d|_OˆOq is compact. Thus, as µ`

OzO˘

“0 we can assert that pO, d|_OˆO, µ|_Oq is a compact doubling metric measure space supporting a weak p1, pq-Poincar´e inequality. In what follows, to simplify the notation we set pY, δ, νq:“ pO, d|_OˆO, µ|_Oq.

Step 1: two auxiliary lemmas. We need the following two lemmas (cf. Lemmas 3.7 and 3.8).

Lemma 3.7. If pą κ then for every r ą0 and ν-a.e. x¯ P Y there exists Cpr,xq ą¯ 0 such that

|upyq ´upzq| ďCpr,xqδpy, zq¯ ^1´κp ˆż

B6σrp¯xq

|∇_νu|^pdν

˙¹

p

for all uPH_ν^1,ppY;R^mq and all y, z P B_rpx¯q, where σě1 is given by (3.2).

Proof of Lemma 3.7. From [Haj03, Theorem 9.7] we can assert that there exists c ą 0 such that

|wpyq ´wpzq| ďcr^κpδpy, zq^1´κp ˆ

´ ż

B6σrp¯xq

g_w^pdν

˙¹

p

(3.7) for all w P H_ν^1,ppYq, all ¯x P Y, all r ą 0 and all y, z P B_rpx¯q, where σ ě1 is given by (3.2) and g_w P L^p_νpYqdenotes the minimal p-weak upper gradient forw. On the other hand, from Remark 3.1 it is easy to see that for every r ą 0 and ν-a.e. ¯x P Y there exists θpr,xq ą¯ 0 such that

νpBrp¯xqq ěθpr,xqr¯ ^κ.

But g_w ď α|D_νw| with α ě 1 (see [Che99, §4]) and so ´ş

B6σrp¯xqg^p_wdν ď α^p´ş

B6σrp¯xq|D_νw|^pdν.

Thus, for every rą0,ν-a.e. ¯xPY and everyy, z P Brp¯xq, (3.7) can be rewritten as follows

|wpyq ´wpzq| ďCpr,xqδpy, zq¯ ^1´κp ˆż

B6σrp¯xq

|D_νw|^pdν

˙¹_p

with Cpr,xq “¯ cα{θpr,xq ą¯ 0. It follows that for every r ą0 and ν-a.e. ¯xPY, we have

|upyq ´upzq| ď Cpr,x¯qδpy, zq^1´

κ p

m

ÿ

i“1

ˆż

B6σrp¯xq

|D_νu_i|^pdν

˙_p¹

ď Cpr,xqδpy, zq¯ ^1´κp

˜ż

B6σrp¯xq m

ÿ

i“1

|D_νu_i|^pdν

¸_p¹

“ Cpr,xqδpy, zq¯ ^1´κp ˆż

B6σrp¯xq

|∇_νu|^pdν

˙¹

p

for all uPH_ν^1,ppY;R^mq and ally, z P B_rp¯xq, and the proof of Lemma 3.7 is complete.

Denote the space of continuous functions from Y to R^m byCpY;R^mq. As a consequence of Lemma 3.7 we have the following result.

Lemma 3.8. If pąκ then H_ν^1,ppY;R^mq continuously embeds into CpY;R^mq, i.e.

H_ν^1,ppY;R^mq ĂCpY;R^mq and there exists K₀ ą0 such that

}u}CpY;R^mq ďK₀}u}_Hν^1,p_pY;Rmq (3.8) for all uPH_ν^1,ppX;R^mq. Moreover, there exists K₁ ą0 such that

|upyq ´upzq| ď K₁δpy, zq^1´κp}∇_νu}_L^p_νpY;Mq (3.9) for all uPH_ν^1,ppY;R^mq and all y, z P Y.

Proof of Lemma 3.8. Applying Lemma 3.7 withr“diampYqand for a fixed ¯x“x₀ PY, where diampYq “suptδpy, zq:y, z P Yu ă 8because pY, δqis compact, we see that

|upyq ´upzq| ď CpdiampYq, x₀qδpy, zq^1´

κ

p}∇_νu}L^pνpY;Mq

ď CpdiampYq, x₀qdiampYq^1´

κ

p}∇_νu}_L^p_νpY;Mq (3.10) for all u P H_ν^1,ppY;R^mq and all y, z P Y. Hence (3.9) holds with K₁ “ CpdiampYq, x₀q and every u P H_ν^1,ppY;R^mq is p1´ ^κ_pq-H¨older continuous. In particular, it follows that H_ν^1,ppY;R^mq Ă CpY;R^mq. On the other hand, given any u P H_ν^1,ppY;R^mq and any y P Y, we have |upyq|^p ď2^pp|upyq ´upzq|^p` |upzq|^pqfor all z P Y, and consequently

νpYq

1

p|upyq| ď2^1`1p ˆż

Y

|upyq ´upzq|^pdνpzq

˙¹_p

`2<sup>1`p1</sup>}u}_L^p_νpY;R^m_q. (3.11)

OMAR ANZA HAFSA AND JEAN-PHILIPPE MANDALLENA

But, by (3.10) we have ˆż

Y

|upyq ´upzq|^pdνpzq

˙¹_p

ďνpYq

1

pCpdiampYq, x₀qdiampYq^1´

κ

p}∇_νu}_L^p_νpY;Mq. (3.12) Hence, combining (3.11) and (3.12) we deduce that for every yPY,

|upyq| ď 2^1`1pCpdiampYq, x₀qdiampYq^1´

κ

p}∇_νu}_L^p_νpY;Mq` 2<sup>1`1p</sup> νpYq

1 p

}u}_L^p_νpY;R^m_q ď K₀}u}_H^1,p

ν pY;R^mq

with K₀ “sup

"

2^1`1pCpdiampYq, x₀qdiampYq^1´

κ p, ^21`

1p

νpYq

1p

*

, and (3.8) follows.

Step 2: end of the proof of Theorem 3.6. As µ` OzO˘

“0, from (3.5) we deduce that

nlim!8}un´u}_L^p_νpY;R^m_q“0 and sup

ně1

}∇νun}L^pνpY;Mq ă 8,

and so sup_ně1}u_n}_Hν^1,p_pY;Rmqă 8. By Lemma 3.8 we can assert that sup_ně1}u_n}CpY;R^mq ă 8, i.e. tununis bounded inCpY;R^mqwithpY, δqa compact metric space. Moreover, using (3.9) we see that tunun is equicontinuous. Consequently, up to a subsequence,

nlim!8}u_n´u}_L⁸_νpY;R^m_q“0 by Arzel`a-Ascoli’s theorem, and (3.6) follows becauseµ`

OzO˘

“0.

3.2. Ru-usc integrands. Let pX, d, µq be a metric measure space, let O Ă X be an open set, letpΩ,F,Pqbe a probability space and letL:OˆMˆΩ!r0,8sbe a Borel measurable stochastic integrand. For eachtap¨, ωquω ĂL¹_µpO;s0,8sqwe define ∆^a_L:r0,1sˆΩ!s´8,8s by

∆^a_Lpτ, ωq:“sup

xPO

sup

ξPLx,ω

Lpx, τ ξ, ωq ´Lpx, ξ, ωq apx, ωq `Lpx, ξ, ωq , where L^x,ω denotes the effective domain of Lpx,¨, ωq.

Definition 3.9. Let ω P Ω. We say that L is radially uniformly upper semicontinuous (ru-usc) at ω if there exists ap¨, ωq PL¹_µpO;s0,8sq such that

lim

τ!1^´

∆^a_Lpτ, ωq ď0.

The concept of ru-usc integrand was introduced in [AH10] and then developed in [AHM11, AHM12a, AHM12b, Man13, AHM14, AHMZ15, AHM18].

Remark 3.10. If Lis ru-usc at ω PΩ then lim_τ!1^´Lpx, τ ξ, ωq ďLpx, ξ, ωqfor all xPO and all ξ P L^x,ω. On the other hand, given ω P Ω, if there exist x P O and ξ P L^x,ω such that Lpx,¨, ωq is lsc at ξ then, for each ap¨, ωq P L¹_µpO;s0,8sq, lim_τ!1´∆^a_Lpτ, ωq ě0, and so if in addition L is ru-usc at ω then lim_τ!1^´∆^a_Lpτ, ωq “0 for some ap¨, ωq PL¹_µpO;s0,8sq.

Remark 3.11. Given ω P Ω, if, for every x P O, Lpx,¨, ωq is convex and 0 P Lx,ω, then L is ru-usc at ω.

The interest of Definition 3.9 comes from the following theorem. (For a proof we refer to [AHM11, Theorem 3.5] and also [AHM12b, §4.2].) Let Lp :O ˆMˆΩ! r0,8s be defined by

Lpx, ξ, ωqp :“ lim

τ!1^´

Lpx, τ ξ, ωq.

Theorem 3.12. Let ωP Ω. If L is ru-usc at ω with ap¨, ωq and if for every xPO, τLx,ω ĂintpLx,ωq for all τ Ps0,1r,

where L^x,ω denotes the effective domain of Lpx,¨, ωq, then:

(i) Lppx, ξ, ωq:“ lim

τ!1^´

Lpx, τ ξ, ωq “

# lim

τ!1^´

Lpx, τ ξ, ωq if ξ PL^x,ω

8 otherwise;

(ii) Lp is ru-usc at ω with ap¨, ωq.

If moreover Lpx,¨, ωq is lsc on intpL^x,ωq then:

(iii) Lpx, ξ, ωq “p

$

&

%

Lpx, ξ, ωq if ξ PintpL^x,ωq lim

τ!1^´Lpx, τ ξ, ωq if ξ P BL^x,ω

8 otherwise;

(iv) for every xPO, Lppx,¨, ωq is the lsc envelope of Lpx,¨, ωq.

The following definition extends Definition 3.9 to a family tL_tutą0 of Borel measurable sto- chastic integrandsL_t:OˆMˆΩ!r0,8s. (WhenL_t “Lfor allt ą0 we retrieve Definition 3.9.)

Definition 3.13. LetωPΩ. We say thattLtutą0 is ru-usc atωif there existstatp¨, ωqutą0 Ă L¹_µpO;s0,8sq, satisfying (2.1) and (2.2), such that

lim

τ!1^´sup

tą0

∆^a_L^t_tpτ, ωq ď0.

For each tą0 and each ρą0, letH_µ^ρLt :OˆMˆΩ!r0,8s be defined by H_µ^ρL_tpx, ξ, ωq:“inf

#

´ ż

Bρpxq

L_tpy, ξ`∇_µwpyq, ωqdµpyq:wPH_µ,0^1,ppB_ρpxq;R^mq +

,

where the space H_µ,0^1,ppB_ρpxq;R^mqis defined as the closure of Lip₀pBρpxq;R^mq:“

!

uPLippO;R^mq:u“0 on OzBρpxq )

with respect to the H_µ^1,p-norm, where LippO;R^mq :“ rLippOqs^m with LippOq denoting the algebra of Lipschitz functions from O to R. Let L₈ :OˆMˆΩ!r0,8s be given by

L₈px, ξ, ωq:“lim

ρ!0lim

t!8H_µ^ρL_tpx, ξ, ωq. (3.13) The following proposition shows that ru-usc is conserved under the operation characterized by (3.13).

Proposition 3.14. Let ω PΩ. If tL_tutą0 is ru-usc at ω withta_tp¨, ωqu_tą0 then L₈ is ru-usc at ω with a₈p¨, ωq given by (2.2).

OMAR ANZA HAFSA AND JEAN-PHILIPPE MANDALLENA

Proof of Proposition 3.14. Fix anyτ P r0,1s, anyxPO and anyξPL8,x,ω, whereL8,x,ω

is the effective domain of L₈px,¨, ωq. Then L₈px, ξ, ωq “ lim_ρ!0lim_t!8H_µ^ρL_tpx, ξ, ωq ă 8 and without loss of generality we can suppose that H_µ^ρLtpx, ξ, ωq ă 8 for all ρ ą0 and all t ą 0. Fix any ρ ą 0 and any t ą 0. By definition, there exists tw_nun Ă H_µ,0^1,ppB_ρpxq;R^mq such that:

H_µ^ρLtpx, ξ, ωq “ lim

n!8´ ż

Bρpxq

Ltpy, ξ`∇µwnpyq, ωqdµpyq; (3.14) ξ`∇_µw_npyq PLt,y,ω for all n ě1 andµ-a.a. yPB_ρpxq, (3.15) where Lt,y,ω denotes the effective domain of L_tpy,¨, ωq. Moreover, for every ně1,

H_µ^ρLtpx, τ ξ, ωq ď ´ ż

Bρpxq

Lt

`y, τpξ`∇µwnpyqq, ω˘ dµpyq

since τ w_n PH_µ,0^1,ppB_ρpxq;R^mq, and so δ^τ_ρ,tpx, ξ, ωq ď lim

n!8

´ ż

Bρpxq

`L<sub>t</sub>py, τpξ`∇_µw_npyqq, ωq ´L_tpy, ξ`∇_µw_npyq, ωq˘

dµpyq (3.16) with δ^τ_ρ,tpx, ξ, ωq:“H_µ^ρL_tpx, τ ξ, ωq ´H_µ^ρL_tpx, ξ, ωq. Taking (3.15) into account, for every ně1 and µ-a.e. y PBρpxq, one has

λ^τ_t,npy, ξ, ωq ď ∆^a_L^t

tpτ, ωq`

a_tpy, ωq `L<sub>t</sub>py, ξ`∇_µw_npyq, ωq˘ , with λ^τ_t,npy, ξ, ωq:“L_t`

y, τpξ`∇_µw_npyqq, ω˘

´L_t`

y, ξ`∇_µw_npyq, ω˘

, hence

´ ż

Bρpxq

λ^τ_t,npy, ξ, ωqdµ ď ∆^a_L^t_tpτ, ωq

˜

´ ż

Bρpxq

a_tpy, ωqdµ` ´ ż

Bρpxq

L_tpy, ξ`∇_µw_npyq, ωqdµ

¸

for all n ě1. Letting n !8 and using (3.14) and (3.16), it follows that δ^τ_ρ,tpx, ξ, ωq ď ∆^a_L^t

tpτ, ωq

˜

´ ż

Bρpxq

a_tpy, ωqdµpyq `H_µ^ρL_tpx, ξ, ωq

¸

ď ∆pτ, ωq

˜

´ ż

Bρpxq

a_tpy, ωqdµpyq `H_µ^ρL_tpx, ξ, ωq

¸

(3.17) for all ρą0 and allt ą0, where ∆pτ, ωq:“sup_są0∆^a_L^s_spτ, ωq. By lettingt !8 and ρ!0 in (3.17), we get

L₈px, τ ξ, ωq ´L₈px, ξ, ωq ď ∆pτ, ωq`

a₈px, ωq `L₈px, ξ, ωq˘ with a₈p¨, ωq PL¹_µpO;s0,8sq given by (2.2), which implies that ∆^a_L⁸

8pτ, ωq ď ∆pτ, ωq for all τ P r0,1s. As tL_tutą0 is ru-usc at ω with ta_tp¨, ωqu_tą0, i.e. lim_τ!1^´∆pτ, ωq ď0, we conclude that lim_τ!1^´∆^a_L⁸

8pτ, ωq ď 0 which means that L₈ is ru-usc at ω with a₈p¨, ωq.

Remark 3.15. In the proof of Proposition 3.14 we do not need (2.1) . In fact, (2.1) will be used in the proof of the Γ-convergence result (see Sect. 4).

3.3. Integral representation of the Vitali envelope of a set function. What follows was first developed in [BFM98, BB00] (see also [AHM16, AHM17, AHCM17, AHM18]). Let pO, dq be a metric space, letOpOq be the class of open subsets of O and letµ be a positive finite Radon measure on O. We begin with the concept of the Vitali envelope of a set function.

For eachεą0 and eachAPOpOq, we denote the class of countable familiestB_i :“B_ρipx_iquiPI

of disjoint open balls ofA withx_i PA, ρ_i Ps0, εrand µpBB_iq “0 such that µpAz Y_iPIB_iq “0 byV_εpAq.

Definition 3.16. GivenS:OpOq!r0,8s, for eachεą0 we defineS^ε:OpOq!r0,8s by S^εpAq:“inf

# ÿ

iPI

SpBiq:tBiuiPI PVεpAq +

.

By the Vitali envelope of S we call the set function S^˚ :OpOq!r´8,8s defined by S^˚pAq:“sup

εą0

S^εpAq “ lim

ε!0S^εpAq.

The interest of Definition 3.16 comes from the following integral representation result. (For a proof we refer to [AHM18, §3.3] or [AHCM17, §A.4].)

Theorem 3.17. Let S:OpOq!r0,8s be a set function satisfying the following two condi- tions:

(i) there exists a finite Radon measureν onΩwhich is absolutely continuous with respect to µ such that SpAq ďνpAq for all APOpOq;

(ii) S is subadditive, i.e. SpAq ď SpBq `SpCq for all A, B, C P OpOq with B, C ĂA, BXC “ H and µpAzBYCq “0.

Then lim_ρ!0 ^SpB_µpB^ρp¨qq

ρp¨qq PL¹_µpOq and for every APOpOq, one has S^˚pAq “

ż

A ρlim!0

SpB_ρpxqq µpB_ρpxqqdµpxq. 4. Proofs

4.1. Proof of the lower bound. Here we prove Proposition 2.6.

Proof of Proposition 2.6. Fixω PΩ. LetuP H_µ^1,ppO;R^mqand let tu_tutą0 ĂH_µ^1,ppO;R^mq be such that }u_t´u}_L^p_µpO;R^m_q !0. We have to prove that

lim

t!8

E_tpu_t, ωq ě ż

O

lim

τ!1^´

ρlim!0 lim

t!8

H_µ^ρL_tpx, τ∇_µupxq, ωqdµpxq. (4.1) Without loss of generality we can assume that lim_t!8E_tpu_t, ωq “lim_t!8E_tpu_t, ωq ă 8, and so

sup

tą0

Etput, ωq ă 8. (4.2)

In particular, sup_tą0}∇_µu_t}L^pµpO;Mq ă 8 because tL_tutą0 is p-coercive, see (C₆). Then

∇µutpxq PGfor all tą0 andµ-a.a. xPO (4.3)