Integral representation and relaxation of local functionals on Cheeger-Sobolev spaces

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Integral representation and relaxation of local functionals on Cheeger-Sobolev spaces

Omar Anza Hafsa, Jean-Philippe Mandallena

To cite this version:

Omar Anza Hafsa, Jean-Philippe Mandallena. Integral representation and relaxation of local func- tionals on Cheeger-Sobolev spaces. 2021. �hal-03270938v2�

INTEGRAL REPRESENTATION AND RELAXATION OF LOCAL FUNCTIONALS ON CHEEGER-SOBOLEV SPACES

OMAR ANZA HAFSA AND JEAN-PHILIPPE MANDALLENA

Abstract. We prove an integral representation theorem for local functionals with poly- nomial growth defined on Cheeger-Sobolev spaces. More precisely, we give a version of the well-known Buttazzo-Dal maso’s integral representation theorem in the framework of Cheeger-Sobolev spaces. The integral representation theorem is used to prove a relaxation theorem.

1. Introduction

LetpX, d, µq be a metric measure space with µa nontrivial Borel (regular) measure on X and pX, dq is a separable metric space. Let p Ps1,8r. Cheeger showed in a seminal paper [Che99] that if we assume thatµis doubling and pX, d, µq enjoys a p1, pq-Poincaré inequality (see Section 2), then X has a measurable differentiable structure. That is there exists a countable collection t`

X_k, γ^k˘

ukPN of measurable sets X_k and of Lipschitz “coor- dinate” functions γ^k :“

´

γ₁^k, . . . , γ_N^k_pkq

¯

: X_k Ñ R^Npkq such that µpXz Y_kX_kq “ 0 and each Lipschitz functionf :X ÑRis differentiable in the sense that there exists a bounded measurable functionD_µ^kf P L⁸_µ `

X_k;R^Npkq˘

such that forµ-a.e.xP X_k, limρÑ0 sup

yPBρpxq

ˇˇfpyq ´fpxq ´@

D^k_µfpxq, γ^kpyq ´γ^kpxqDˇ ˇ

ρ “0.

The Cheeger-Sobolev spaceH_µ^1,ppX;R^mqonX, introduced by Cheeger [Che99] (see Sec- tion 2, Definition 3), can be defined as the completion of Lipschitz functions.

Our goal is to show an integral representation and a relaxation theorem for local func- tionals withp-growth defined on Cheeger-Sobolev spacesH_µ^1,ppΩ;R^mqwhereΩĂX is an open set with finite measure. More precisely, we want to give a version of the integral rep- resentation and relaxation results of [BDM85, BFLM02] in the setting of Cheeger-Sobolev spaces.

The proof of the integral representation of [BDM85] on open sets of Euclidean spaces splits into several steps, first, the integrand is defined on linear functions, which allows easily to write an integral representation on continuous piecewise affine functions. Then it is shown that necessarily the integrand is continuous with respect to the second vari- able (Carathéodory integrand) by proving the “zig-zag lemma” whose proof uses the lower semicontinuity property along specific construction of continuous piecewise affine func- tions. The conclusion comes by passing to the limit and by using the local approximation of Sobolev functions by continuous piecewise affine functions and the continuity (and the growth conditions) with respect to the second variable of the integrand. At first glance, there does not seem to be an easy way to adapt this strategy in Cheeger-Sobolev spaces.

Université de Nîmes, Laboratoire MIPA, Site des Carmes, Place Gabriel Péri, 30021 Nîmes, France E-mail addresses:<[email protected]>, <[email protected]>. Key words and phrases. Integral representation, Relaxation, Integral functionals defined on Cheeger-Sobolev spaces.

1

Especially, we do not know how to adjust the zig-zag lemma to obtain the continuity of the integrand. One way is to assume a convexity condition on the functional (see [MPSC20, MV20]).

The integral representation result of [BFLM02, Theorem 2, pp. 189] shows that the integrand can be written as limit, when the radius of balls goes to zero, of the average of minimization Dirichlet problems associated with the functional on small balls. The strat- egy of the proof, known as the “global relaxation method”, uses mainly an intermediate representation result of an envelope, similar to the Carathéodory construction in mea- sure theory (see Subsection 6.3), of local minimization Dirichlet problems associated with the functional. The advantage of the method is that it avoids the use of approximation by continuous piecewise affine functions. It can therefore be adapted more easily to the framework of Cheeger-Sobolev spaces, we already got several results by following this path, see [AHM15, AHM17, AHM18]. We must emphasize that this strategy makes significant use of the coercivity of the functional, which is not the case of the Buttazzo and Dal Maso’s integral representation theorem [BDM85].

One motivation, for developing the calculus of variations in the setting of metric mea- sure spaces, comes from applications to hyperelasticity. In fact, the interest of considering a general measure is that its support can be interpreted as a hyperelastic structure 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. (We refer the reader to [BBS97, Zhi02, CJLP02]and [CPS07, Chapter 2, §10] and the references therein). Another motivation is the development of the cal- culus 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 satisfying generalized Ricci bounds (see [KM16] for more details). In- deed, all these spaces are examples of doubling metric measure spaces satisfying a Poincaré inequality on which our integral representation and relaxation results on Cheeger-Sobolev spaces could be applied.

We assume in the following of the paper that µ is doubling, pX, d, µq enjoys a p1, pq- Poincaré inequality,pX, dqis a complete separable metric space, andpX, d, µqsatisfies the annular decay property (see Definition 4).

Throughout the rest of the paperΩĂXdenotes an open set of finite measureµpΩq ă 8. We denote byOpΩqthe class of all open subsets ofΩ.

Our first result is an integral representation theorem in Cheeger-Sobolev spaces:

Theorem 1. LetF :H_µ^1,ppΩ;R^mq ˆOpΩq Ñ r0,8ssatisfy

(C₁) for every u P H_µ^1,ppΩ;R^mq the set functionF pu,¨q is the restriction to OpΩq of a positive Radon measure;

(C₂) F p¨, Oq is local, i.e. F pu, Oq “ F pv, Oq whenever u “ v µ-a.e. in O for all pu, vq P H_µ^1,ppΩ;R^mq² and allO POpΩq;

(C₃) F pu`z, Oq “F pu, Oqfor allz PR^m, alluPH_µ^1,ppΩ;R^mqand allO POpΩq;

(C₄) there existcą0,b ě0andaP L¹_µpΩqsuch that for everypu, Oq PH_µ^1,ppΩ;R^mq ˆOpΩq c

ˆ

O

|∇_µupxq|^pdµpxq ď Fpu, Oq ď ˆ

O

apxq `b|∇_µupxq|^pdµpxq where∇_µuis theµ-gradient ofu.

2

(C₅) for everyO POpΩqthe functionalFp¨, OqisL^p_µ-lower semicontinuous, i.e. for everyu,tu_nunPN Ă L^p_µpΩ;R^mqsatisfyinglim_nÑ8}u_n´u}_L^p_µpΩ;Rmq “0we have

lim

nÑ8

Fpu_n, Oq ě F pu, Oq.

Then there exists a Borel measurable functionf : ΩˆMÑ r0,8ssuch that (i) for everyO POpΩqand everyuPH_µ^1,ppΩ;R^mq

F pu, Oq “ ˆ

O

fpx,∇µupxqqdµpxq; (1) (ii) for everyk PN, forµ-a.e.xP ΩXX_k and everyξPM

fpx, ξq:“ lim

ρÑ0 inf

ϕPH_µ,0^1,ppBρpxq;R^mq

F `

ξ¨γ^kp¨q `ϕ, B_ρpxq˘ µpB_ρpxqq ; (iii) the functionf isH_µ^1,p-quasiconvex, i.e. for everyξPMand forµ-a.e.xPΩ

fpx, ξq “ lim

ρÑ0 inf

ϕPH_µ,0^1,ppBρpxq;R^mq Bρpxq

fpy, ξ`∇_µϕpyqqdµpyq; (iv) forµ-a.e.xPΩand for everyξP Mwe have

c|ξ|^p ďfpx, ξq ďapxq `b|ξ|^p wherecą0,b ě0anda PL¹_µpΩqare given by(C4);

(v) if there exists a Borel measurable functionfr: ΩˆMÑ r0,8ssuch that for everyO POpΩq and everyuPH_µ^1,ppΩ;R^mq

Fpu, Oq “ ˆ

O

frpx,∇_µupxqqdµpxq (2) then forµ-a.e.xPX and for everyξ PM

frpx, ξq “fpx, ξq.

When the functional is not necessarilyL^p_µ-lower semicontinuous we need to consider the L^p_µ-lower semicontinuous envelope ofF p¨, Oqdefined by

H_µ^1,ppΩ;R^mq QuÞÝÑF pu, Oq:“inf

"

lim

nÑ8

F pu_n, Oq:u_n ÑuinL^p_µpΩ;R^mq

* . We have the following relaxation theorem:

Theorem 2. LetF :H_µ^1,ppΩ;R^mq ˆOpΩq Ñ r0,8ssatisfying(C₁)-(C₄). Then there exists a Borel measurable functionf : ΩˆMÑ r0,8ssuch that

(i) for everyO POpΩqand everyuPH_µ^1,ppΩ;R^mq F pu, Oq “

ˆ

O

fpx,∇µupxqqdµpxq; (3) (ii) for everyk PN, forµ-a.e.xP X_kand for everyξP M

fpx, ξq:“ lim

ρÑ0 inf

ϕPH_µ,0^1,ppBρpxq;R^mq

F `

ξ¨γ^kp¨q `ϕ, B_ρpxq˘ µpB_ρpxqq ; (iii) the functionf isH_µ^1,p-quasiconvex;

3

(iv) forµ-a.e.xPΩand for everyξP Mwe have

c|ξ|^p ďfpx, ξq ďapxq `b|ξ|^p wherecą0,b ě0anda PL¹_µpΩqare given by(C₄).

The following consequence of the relaxation Theorem 2 is a characterization of the lower semicontinuity of integral functionals. This is an improvement of theH_µ^1,p-quasiconvexity, a necessary condition (playing the same role as the quasiconvexity concept in the Euclidean case, see for instance [BM84]) studied in [AHM20] (see Subsection 6.2).

Corollary 1. Letf : ΩˆMÝÑ r0,8sbe a Borel measurable function. Assume that there exists cą0,bě0andaPL¹_µpΩqsuch that forµ-a.e.xP Ωand for everyξ PMwe have

c|ξ|^p ďfpx, ξq ď apxq `b|ξ|^p. The following two assertions are equivalent:

(i) forµ-a.e.xPΩand for everyξP Mit holds fpx, ξq “ lim

ρÑ0 inf

ϕPH_µ,0^1,ppBρpxq;R^mq Bρpxq

fpy, ξ`∇µϕqdµ.

(ii) for everyO POpΩq, the functional H_µ^1,ppO;R^mq QuÞÝÑ

ˆ

O

fpx,∇_µupxqqdµpxq is L^p_µ-lower semicontinuous.

The plan of the paper is as follows. In Section 2 we provide the materials about metric measure spaces and Cheeger-Sobolev spaces we need for our purposes. In Section 3 we give the proof of Theorem 1. The proof splits into several steps, we use first Lemma 10 which provides an integral representation of the Vitali envelope of the local minimization Dirichlet problem associated with the functional, then we can localize, using cut-off techniques, the formula of the integrand by replacing u withu_xp¨q :“ upxq ` ∇<sub>µ</sub>upxq ¨`

γ^kp¨q ´γ^kpxq˘ . In the last step, we show that the integrand is Borel measurable. In Section 4, we prove the relaxation theorem Theorem 2 with the help of Theorem 1 and by using mainly the De Giorgi-Letta Lemma which gives sufficient conditions for increasing set functions on open sets to be a measure. In Section 5 we prove Corollary 1 which is a consequence of Theorem 2 and Corollary 2 about the equality of Borel measurable integrands. This is a recast in the setting of metric measure space of an Alberti’s result [Alb91]. The last Section 6 is devoted to the auxiliary results we need in the proofs.

Notation.

‚ We will denote byBpΩqthe Borelσ-algebra ofXandB_µpΩqtheµ-completion ofBpΩq.

‚ We will denote byB_ρpxq:“ ty P X : dpx, yq ăρuthe open ball, and byB_ρpxq:“ ty P X :dpx, yq ď ρuthe closed ball, centered atxwith radiusρą0.

‚ For every measurable setAĂΩwith positive measure, and for every nonnegative mea- surable or integrable functionf onA, we set

A

f dµ :“ 1 µpAq

ˆ

A

fpxqdµpxq.

‚ The algebra of Lipschitz functions fromΩtoRis denoted byLippΩq.

4

2. Preliminaries: the metric measure spaces, the Cheeger–Sobolev spaces

Letpą1be a real number, letpX, d, µqbe a metric measure space, whereµis a nontrivial locally finite Borel regular measure on X and pX, dq is a separable metric space. In what follows, we assume thatµisdoubling, i.e. there exists a constantC_d(called doubling constant) such that

@xPX @ρą0 µpB_2ρpxqq ď C_dµpB_ρpxqq. (4) The concept of upper gradient was introduced by Heinonen and Koskela (see [HK98]), and generalized by Cheeger (see [Che99, Definition 2.8]):

Definition1.

(i) A Borel functiong :X Ñ r0,8sis said to be anupper gradientforf :X ÑRif

|fpcp1qq ´fpcp0qq| ď ˆ 1

0

gpcpsqqds for all continuous rectifiable curvesc:r0,1s ÑX.

(ii) A functiong PL^p_µpXqis said to be ap-weak upper gradientforf PL^p_µpXqif there exist tfnun ĂL^p_µpXqandtgnunĂL^p_µpXqsuch that for eachně1,gnis an upper gradient forf_n,f_n Ñf inL^p_µpXqandg_nÑginL^p_µpXq.

Definition2. The metric measure spacepX, d, µqenjoys ap1, pq-Poincaré inequality withp P s1,8rif there existC_p ą0andσ ě1such that for everyxP Xand everyρą0,

Bρpxq

ˇ ˇ ˇ ˇ ˇ

fpyq ´

Bρpxq

f dµ ˇ ˇ ˇ ˇ ˇ

dµpyq ď ρC_p

˜

Bσρpxq

g^pdµ

¸¹

p

(5) for everyf PL^p_µpXqand everyp-weak upper gradientg PL^p_µpXqforf.

From Cheeger [Che99, Theorem 4.38] (see also Keith [Kei04, Definition 2.1.1 and The- orem 2.3.1]) we have:

Theorem 3. Ifµis doubling, i.e.(4)holds, andXenjoys ap1, pq-Poincaré inequality, i.e.(5)holds, then there exist a countable family t`

X_k, γ^k˘

u_kPN of µ-measurable disjoint subsets X_k of X with µpXz Y⁸_k“0 X_kq “ 0and of functionsγ^k “

´

γ₁^k,¨ ¨ ¨, γ_Npkq^k

¯

: X ÑR^Npkq withγ_i^k P LippXq satisfying the following properties:

(i) there exists an integerN ě1such thatNpkq P t1, . . . , Nufor allk PN; (ii) for everyk P Nand everyf P LippXqthere is a uniqueD^k_µf P L⁸_µ `

X_k;R^Npkq˘

such that forµ-a.e.xPXk,

ρÑ0lim sup

yPBρpxq

ˇ

ˇfpyq ´fpxq ´@

D_µ^kfpxq, γ^kpyq ´γ^kpxqDˇ ˇ

ρ “0, (6)

wheref_x PLippXqis given byf_xpyq:“fpxq `@

D^k_µfpxq, γ^kpyq ´γ^kpxqD

; in particular D^k_µf_xpyq “D^k_µfpxq forµ-a.a. yP X_k;

(iii) the operatorD_µ: LippXq ÑL⁸_µ `

X;R^N˘

given by D_µf :“ ÿ

kPN

1XkD_µ^kf,

where 1Xk denotes the characteristic function ofX_k, is linear and, for each f, g P LippXq, we have

D_µpf gq “f D_µg`gD_µf;

5

(iv) for everyf PLippXq,D_µf “0µ-a.e. on everyµ-measurable set wheref is constant.

We setM “R^mˆN whereN is given by Theorem 3 (i). For every k P Nwe denote by

| ¨ |k the Euclidean norm on R^mˆNpkq whereNpkq is given by Theorem 3, and we set for everyxPΩand everyξ PM

|ξ|_x :“

8

ÿ

k“0

|ξ|_k1Xkpxq.

There exists Borel setsΩĄB_k ĄX_kXΩsuch thatµpΩz pY⁸_k“0B_kqq “0since the regularity ofµ. If we defineN : ΩˆMÑR`by

Npx, ξq:“

8

ÿ

k“0

|ξ|k1Bkpxq,

then we can see thatN is Borel measurable and coincide with| ¨ |x forµ-a.e.xPΩ.

Lemma 1. There exists a Borel measurable functionN : ΩˆMÑR` such that forµ-a.e. inΩ and for everyξ PMwe haveNpx, ξq “ |ξ|_x.

In the following, for everyk P Nand everyxP X_kXΩ, we simply denote the norm of ξ PR^Npkqˆm ĂMby|ξ|instead of|ξ|_x.

LetLippΩ;R^mq:“ rLippΩqs^m and let∇µ : LippΩ;R^mq ÝÑL⁸_µ pΩ;Mqgiven by

∇_µu:“

¨

˝ D_µu₁

... D_µu_m

˛

‚withu“ pu₁,¨ ¨ ¨, u_mq.

From Theorem 3 (iii) we see that for everyuPLippΩ;R^mqand everyf PLippΩq, we have

∇µpf uq “ f∇µu`Dµf bu. (7) Definition 3. The p-Cheeger–Sobolev space H_µ^1,ppΩ;R^mq is defined as the completion of the space of Lipschitz functionsLippΩ;R^mqwith respect to the norm

}u}_H^1,p

µ pΩ;R^mq:“ }u}_L^p_µpΩ;R^m_q` }∇µu}_L^p_µpΩ;Mq. (8) Taking Proposition 1 (ii) below into account, since}∇_µu}_L^p_µpΩ;Mq ď }u}_H^1,p

µ pΩ;R^mq for all u P LippΩ;R^mqthe linear map∇_µ fromLippΩ;R^mqtoL^p_µpΩ;Mq has a unique extension toH_µ^1,ppΩ;R^mqwhich will still be denoted by∇_µand will be called theµ-gradient.

Remark1. In fact, the original Cheeger’s definition [Che99, (2.1), pp. 440] of H_µ^1,ppΩqis all the functionsuPL^p_µpΩqfor which|u|1,pă 8where

|u|_1,p “ |u|_L^p_µpΩq`inf

"

lim

nÑ8

|g_un|_L^p_µpΩq:L^p_µpΩq Qu_n ÑuinL^p_µ

*

whereg_un is an upper gradient foru_nfor alln P N. Cheeger [Che99, Theorem 4.47, pp.

459] shows that the spaceH_µ^1,ppΩqcan be seen as the completion of Lipschitz functions for the norm |u|_L^p_µpΩq ` |D_µu|_L^p_µpΩq for all u P LippΩq. This latter norm coincide with | ¨ |1,p

by combining [Che99, Definition 2.9 and Theorem 2.10, pp. 441] and [Che99, Corollary 4.41, pp. 458].

For more details on the various possible extensions of the 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 gathers some results of many authors and provides useful properties for dealing with calculus of variations in the metric measure setting.

6

Proposition 1. Under the hypotheses of Theorem 3, we have:

(i) (from [HKST15, Theorem 3.4.3, pp. 73]) the metric measure spaceX satisfies the Vi- tali covering theorem, i.e. for every A Ă X and every family F of closed balls in X, if inf ρą0 :B_ρpxq PF(

“ 0for allx P Athen there exists a countable disjoint subfamily G ofF such thatµpAz YBPGBq “0; in other words,AĂ pYBPGBq YN withµpNq “0;

(ii) (from [FHK99, Theorem 10]) the µ-gradient is closable in H_µ^1,ppΩ;R^mq, i.e. for every u P H_µ^1,ppΩ;R^mq and every measurable set A Ă Ω, if upxq “ 0 for µ-a.a. x P A then

∇_µupxq “ 0forµ-a.a. xP A;

(iii) (from [BB11, Corollary 4.24 pp. 93], [BB11, Theorem 5.51, pp. 142] and [Che99,

§4])the metric spaceX enjoys ap-Sobolev inequality, i.e. there existsCS ą0such that

˜ˆ

Bρpxq

|v|^pdµ

¸_p¹

ďρC_S

˜ˆ

Bρpxq

|∇_µv|^pdµ

¸¹_p

(9) for all0ăρďρ₀, withρ₀ ą0, and allv P H_µ,0^1,ppB_ρpxq;R^mq, where, for eachO POpXq, H_µ,0^1,ppO;R^mqis the closure ofLip₀pO;R^mqwith respect toH_µ^1,p-norm defined in(8)with

Lip₀pO;R^mq:“ uPLippX;R^mq:u“0onXzO(

;

(iv) (from [Bjö00, Theorem 4.5 and Corollary 4.6] or [GH13, Theorem 2.12])for every u P H_µ^1,ppΩ;R^mq and µ-a.e. x P Ω there exists u_x P H_µ^1,ppΩ;R^mq given by u_xpyq :“

upxq `∇µupxq ¨`

γ^kpyq ´γ^kpxq˘

such that

∇_µu_xpyq “∇_µupxq forµ-a.a. yPX;

limρÑ0

1 ρ

˜

Bρpxq

|upyq ´u_xpyq|^pdµpyq

¸¹_p

“0; (10)

(v) (from [Che99, Theorem 6.1])for everyxPΩ, everyρą0and everyτ Ps0,1rthere exists a functionϕPLippΩ;r0,1sqsuch that

ϕpxq “0 for allxPΩzB_ρpxq, ϕpxq “1 for allxP B_{τ ρ}pxq and

}D_µϕ}L⁸_µpΩ;R^Nqď C₀

ρp1´τq for someC₀ ą0.

Such aϕis called an Urysohn function for the pair`

ΩzB_ρpxq, B_{τ ρ}pxq˘ .

Definition4. We say that the metric measure spacepX, d, µqsatisfiesthe annular decay property if there existηą0andK₀ ě1such that for everyxPX, everyρą0and everyτ Ps0,1r,

µpBρpxq zBτ ρpxqq ďK0p1´τq^ηµpBρpxqq.

The annular decay property was introduced independently by [Buc99, pp. 521 and §2 pp. 524] and [CM98]. This property holds, for instance, when the metric space is a length space, i.e. metric space in which the distance between points is the infimum of lenghts of rectifiable paths joining those points, see [Buc99, Corollary 2.2], [CM98, Lemma 3.3], [Che99, Proposition 6.12] and [HKST15, Proposition 11.5.3, pp. 328].

Remark2. We can remark that, when the annular decay property holds, the boundary of balls is of zero measure, indeed, ifxP X,ρą0andτ Ps¹₂,1r, we have

µpBB_ρpxqq ďµ

´ B^ρ

τ pxq zB_ρpxq

¯

ďK₀p1´τq^ηµ

´ B^ρ

τ pxq

¯

ďK₀p1´τq^ηµpB_2ρpxqq, lettingτ Ñ1we obtain thatµpBB_ρpxqq “ 0.

7

Lemma 2. Assume that the annular decay property holds. Letλbe a positive Radon measure. Let xPX satisfy

limρÑ0

λpB_ρpxqq

µpB_ρpxqq “: dλ

dµpxq ă 8. (11)

Then for everyτ Ps0,1r

ρÑ0lim

λpB_ρpxq zB_{τ ρ}pxqq

µpB_ρpxqq ďK₀p1´τq^η dλ

dµpxq. (12)

In particular, we have

τÑ1limlim

ρÑ0

λpB_ρpxq zB_{τ ρ}pxqq µpBρpxqq “0.

Proof of Lemma 2. LetxP Xsatisfy (11). We can write for everyρą0 λpB_ρpxq zB_{τ ρ}pxqq

µpB_ρpxqq “ λpB_ρpxqq

µpB_ρpxqq´ µpB_{τ ρ}pxqq µpB_ρpxqq

λpB_{τ ρ}pxqq µpB_{τ ρ}pxqq

“

ˆλpB_ρpxqq

µpB_ρpxqq´ λpB_{τ ρ}pxqq µpB_{τ ρ}pxqq

˙

` µpB_ρpxq zB_{τ ρ}pxqq µpB_ρpxqq

λpB_{τ ρ}pxqq µpB_{τ ρ}pxqq ď

ˆλpB_ρpxqq

µpB_ρpxqq´ λpB_{τ ρ}pxqq µpB_{τ ρ}pxqq

˙

`K₀p1´τq^η λpB_{τ ρ}pxqq µpB_{τ ρ}pxqq, lettingρÑ0and using (11), we obtain (12).

We will need a version of the Rellich-Kondrachov theorem in the metric measure spaces setting. The following result is due to [HK00, Theorem 8.1, pp. 37] (see also [Kał99]).

Theorem 4. LetB ĂΩa ball. LettϕnunPNĂH_µ^1,ppB;R^mqbe a bounded sequence, i.e.

sup

nPN

ˆˆ

B

|ϕ_n|^pdµ

˙¹_p

` ˆˆ

B

|∇_µϕ_n|^pdµ

˙¹_p ă 8.

Then there exist a subsequencetϕ_niuiPNĂH_µ^1,ppB;R^mqandϕPH_µ^1,ppB;R^mqsuch that

iÑ8lim ˆ

B

|ϕ_ni´ϕ|^pdµ“0.

Outline of the proof of Theorem 4. For the sake of simplicity we assume that m “ 1. By a development of Cheeger [Che99, pp. 449-450], we have, in our setting (doubling measure and Poincaré inequality), for everyuP H_µ^1,ppXq, everyxPXand everyrą0the following Poincaré-Sobolev inequality

ˆ

Brpxq

ˇ ˇ ˇ ˇ

upyq ´

Brpxq

udµ ˇ ˇ ˇ ˇ

χp

dµpyq

˙_χp¹ ďCr

ˆ

Brpxq

|Dµu|^pdµpyq

˙¹_p

(13) for someχ ą 1 which only depends on the doubling constantC_d in (4) and the constant C_p in (5), and for some C ą 0 depending on the constantsC_p, C_d and on the constant σ appearing in (5).

Consider a ballB “B_ρpxq ĂΩand a bounded sequencetϕ_nunPNĂH_µ^1,ppB;R^mq.

From (13), we have for everynP N

›

›

›

› ϕ_n´

B

ϕ_ndµ

›

›

›

›L^χpµ pBq

ďCµpBq

1 χp´_p¹

ρ}D_µϕ_n}_L^p

µpBq 8

thus

}ϕ_n}_L^χp_{µ pBq} ďµpBq^χp1 ˇ ˇ ˇ ˇ _B

ϕ_ndµ ˇ ˇ ˇ

ˇ`CµpBq^{χp1 ´1p}ρ}D_µϕ_n}_L^p

µpBq, and by using the Hölder inequality we obtain

}ϕ_n}_L^χp_{µ pBq} ďµpBq^{χp1 ´1p}

´

}ϕ_n}_L^p_µpBq`Cρ}D_µϕ_n}_L^p

µpBq

¯

which means that the sequencetϕ_nunPNĂL^χp_µ pBqis bounded. Thus, there exists a subse- quence (not relabelled) which weakly converges inL^χp_µ pBqto someϕPL^χp_µ pBq.

Lemma 3. [HK00, Lemma 8.2, pp. 37]Let B¹ Ă B be a ball and χ ą 1. Lettψ_nunPN Ă L^χp_µ pB¹qbe a bounded sequence. Iftψ_nu_nPNconverges in measure toψ PL^χp_µ pB¹qthen

nÑ8lim }ψ_n´ψ}L^pµpB¹q “0.

Taking Lemma 3 into account, we see that it suffices to show thattϕ_nu_nPN converges in measure toϕ.

Lett Ps0,1r. Fixε ą 0andn P N. We set for every positiver ă ^ρp1´tq_σ (whereσ is the constant appearing in (5)) and everyxPtB :“B_tρpxq

ϕ_rpxq:“

Brpxq

ϕdµ and ϕ_n,rpxq:“

Brpxq

ϕ_ndµ.

We have

µptBX r|ϕ_n´ϕ| ąεsq ďµ

´ tBX

”

|ϕ_n´ϕ_n,r| ą ε 3

ı¯

`µ

´ tBX

”

|ϕ_n,r´ϕ_r| ą ε 3

ı¯

`µ

´ tBX

”

|ϕ_r´ϕ| ą ε 3

ı¯

Since the Lebesgue differentiation theorem, µ` tBX“

|ϕ_r´ϕ| ą ^ε₃‰˘

goes to0 asr Ñ 0. The term µ`

tBX“

|ϕ_n,r´ϕ_r| ą ^ε₃‰˘

tends to 0 as n Ñ 8 for all r ą 0, since the weak convergence oftϕ_nunPNinL^χp_µ pBqtoϕ. Using [HK00, Proof of theorem 3.2, pp. 13-14]

we can deduce for everyxPtB

|ϕ_npxq ´ϕ_n,rpxq| ďKr

˜ sup

ρPs0,σrr Bρpxq

|D_µϕ_n|^pdµ

¸¹

p

ďKr ˆ

sup

rą0 BrpxqXB

|D_µϕ_n|^pdµ

˙¹_p

for some constant K ą 0 depending only on the doubling constant and the constant Cp

appearing in the Poincaré inequality (5). Using the maximal theorem [HK00, Theorem 14.13], there existsC¹ ą0depending on the doubling constant only such that

µ

´ tBX

”

|ϕ_n´ϕ_n,r| ą ε 3

ı¯

ďµ ˆ„

sup

rą0 Brp‚qXB

|D_µϕ_n|^pdµą ˆ 1

3K

˙p

´ε r

¯p˙

ďC¹p3Kq^p

´r ε

¯p

sup

nPN

ˆ

B

|D_µϕ_n|^pdµ, thereforesup_nPNµ`

tBX“

|ϕ_n´ϕ_n,r| ą ^ε₃‰˘

tends to0asrÑ0. It follows that

nÑ8lim µptBX r|ϕn´ϕ| ąεsq “0.

Applying Lemma 3 withψ_n “ϕ_nttB,ψ “ϕt_tB andB¹ “tB, we obtain

nÑ8lim }ϕ_n´ϕ}_L^p

µptBq “0.

9

Now, since the sequencetϕ_nunPNis bounded inL^χp_µ pBq, we havesup_nPN}ϕ_n´ϕ}_L^χp_{µ pBq} ă 8. Choose t_ε Ps0,1r such that µpBzt_εBq ď ε^χ´1χ

´

1`sup_nPN}ϕ_n´ϕ}^p_L^χp

µ pBq

¯´1

, by the Hölder inequality we haveˆ

B

|ϕ_n´ϕ|^pdµ“ ˆ

tεB

|ϕ_n´ϕ|^pdµ` ˆ

BztεB

|ϕ_n´ϕ|^pdµ ď }ϕ_n´ϕ}^p_Lp

µptεBq` }ϕ_n´ϕ}^p_Lχp

µ pBqµpBzt_εBq^χ´1χ ď }ϕ_n´ϕ}^p_L^p

µptεBq`ε

lettingn Ñ 8and thenεÑ0we obtain the desired result.

3. Proof of Theorem 1

For a functionalF :H_µ^1,ppΩ;R^mqˆOpΩq Ñ r0,8swe definem:H_µ^1,ppΩ;R^mqˆOpΩq Ñ r0,8sby

mpu, Oq:“inf Fpu`ϕ, Oq:ϕP H_µ,0^1,ppO;R^mq( . 3.1. Proof of (i) and (ii).

Step 1: integral representation ofF pu,¨qviampu,¨q. In this step, we show that for every uPH_µ^1,ppΩ;R^mqand everyO POpΩq

F pu, Oq “ ˆ

O

limρÑ0

mpu, B_ρpxqq

µpB_ρpxqq dµpxq. Letpu, Oq PH_µ^1,ppΩ;R^mq ˆOpΩq. By lemma 10 we have

m^˚_´pu, Oq “ ˆ

O

limρÑ0

mpu, B_ρpxqq

µpB_ρpxqq dµpxq, where

m^˚_´pu, Oq:“sup

εą0

inf

# ÿ

iPI

mpu, B_iq:tB_iu_iPI P V^εpOq +

with for everyεą0 V^εpOq:“

!

tB_iu_iPI ĂBpΩq:I is countable,µ

´ Oz Y

iPIB_i

¯

“0, B_i ĂO,

0ădiampB_iq ďε and B_iXB_j “ H for all i­“j )

. By (C₁) and (C₄), we see thatF pu,¨qis a positive Radon measure which is absolutely con- tinuous with respect toµ, so, we have

m^˚_´pu, Oq ďF pu, Oq. (14)

It remains to prove that

F pu, Oq ďm^˚_´pu, Oq. (15)

Fixε ą0. There exists a countable family of mutually disjoints ballstB_iuiPI PV_εpOqsuch

that ÿ

iPI

mpu, B_iq ď m^˚_´pu, Oq ` ε

2. (16)

Given anyiPI, by definition ofmpu, B_iq, there existsϕⁱ_εP H_µ,0^1,ppB_i;R^mqsuch that F `

u`ϕⁱ_ε, Bi

˘ďmpu, Biq `εµpBiq

2µpOq. (17)

10

Defineϕ_ε : ΩÑR^mby ϕ_ε :“ř8

i“0ϕⁱ_ε1Bi P H_µ,0^1,ppO;R^mq, that is ϕ_ε:“

"

0 inΩzO ϕⁱ_ε inB_i.

Take the sum over the countable family of mutually disjoints ballstB_iuiPIin (17), we obtain by using (C₂) and (C₁)

F pu`ϕ_ε, Oq “ ÿ

iPI

F pu`ϕ_ε, B_iq “ ÿ

iPI

F `

u`ϕⁱ_ε, B_i˘ ďÿ

iPI

mpu, B_iq`ε

2 ďm^˚_´pu, Oq`ε.

If lim_εÑ0}ϕ_ε}L^pµpX;R^mq “ 0 then we get (16) by using (C₁) the lower semicontinuity of F p¨, Oq. So, it remains to prove thatϕ_ε Ñ0inL^p_µpΩ;R^mqasεÑ0. We have by using the Sobolev inequality (9), the coercivity condition (C₄), (14) and the growth condition (C₄)

ˆ

X

|ϕ_ε|^pdµ“ ˆ

O

|ϕ_ε|^pdµ“ÿ

iPI

ˆ

Bi

ˇ ˇϕⁱ_kˇ

ˇ

pdµďÿ

iPI

ε^pC_S^p ˆ

Bi

ˇ ˇ∇_µϕⁱ_εˇ

ˇ

pdµ

ď 2^p´1C_S^pε^p c

ÿ

iPI

F `

u`ϕⁱ_ε, B_i˘

`Fpu, B_iq

ď 2^p´1C_S^pε^p c

`m<sup>˚</sup><sub>´</sub>pu, Oq `ε`F pu, Oq˘ ď 2^pC_S^pε^p

c pε`F pu, Oqq ď 2^pC_S^pε^p

c ˆ

ε` ˆ

O

apxq `b|∇_µupxq|^pdµpxq

˙

by passing to the limitε Ñ0we find thatϕ_ε Ñ0inL^p_µpX;R^mq. Thus we obtain F pu, Oq “ m^˚_´pu, Oq “

ˆ

O

limρÑ0

mpu, B_ρpxqq

µpB_ρpxqq dµpxq.

Step 2: refinement of the formula for the integrand. In this step we show that for µ-a.e.

xPΩ

ρÑ0lim

mpu, B_ρpxqq µpB_ρpxqq “lim

ρÑ0

mpu_x, B_ρpxqq µpB_ρpxqq whereu_x is given by Proposition 1 (iv).

LetxPΩsatisfy

ρÑ0lim Bρpxq

apyqdµpyq “ apxq ă 8; (18)

ρÑ0lim Bρpxq

|∇µu|^pdµ“ |∇µupxq |^p ă 8; (19)

∇_µu_xpyq “ ∇_µupxq forµ-a.e.yP O; (20)

ρÑ0lim 1 ρ

˜

Bρpxq

|upyq ´u_xpyq|^pdµpyq

¸¹

p

“0. (21)

Step 2.1: we prove that lim_ρÑ0 ^mpu_µpB^x,Bρpxqq

ρpxqq ď lim_rÑ0 ^mpu,B_µpB ^rpxqq

rpxqq for µ-a.e. x P Ω. Let ε ą 0, ρą0andtPs0,1r. There existsv P u`H_µ,0^1,ppB_tρpxq;R^mqsuch that

F pv, B_tρpxqq ď εµpB_tρpxqq `mpu, B_tρpxqq. (22)

11

Letτ Pst,1r. We consider a Lipschitz functionϕ: ΩÑ r0,1swhich is a Urysohn function for the pair`

ΩzB_{τ ρ}pxq, B_tρpxq˘

, i.e. satisfyingϕ”1onB_tρpxq,ϕ”0onΩzB_{τ ρ}pxqand }Dµϕ}_L8

µ ď C₀

ρpτ´tq for someC₀ ą0not depending onρ, τ andt. We set

w:“ϕv` p1´ϕqu<sub>x</sub>P u<sub>x</sub>`H_µ,0^1,ppB_ρpxq;R^mq verifying

w “

$

&

%

v inB_tρpxq

ϕu` p1´ϕqu_x inB_{τ ρ}pxq zB_tρpxq u_x inB_ρpxq zB_{τ ρ}pxq.

As in [DM93, pp. 182, Proposition 15.23 and pp. 172, Proposition 14.23] we set

F^˚pw, Bq:“inftFpw, Oq:OpΩq QO ĄBu (23) for all Borel setB ĂΩand allwP H_µ^1,ppΩ;R^mq. The functionalF^˚pw,¨qis a nonnegative Borel measure which extends to all Borel sets the measure F pw,¨q, moreover, since the growth condition (C₄) we have

F^˚pw, Bq ď ˆ

B

apxq `b|∇_µwpxq|^pdµpxq (24) for all Borel setB ĂΩand allwPH_µ^1,ppΩ;R^mq. Since the locality hypothesis (C₂) and (24)

mpu_x, B_ρpxqq ď F pw, B_ρpxqq “F pw, B_tρpxqq `F<sup>˚</sup>pw, B<sub>ρ</sub>pxq zB<sub>tρ</sub>pxqq ďF pv, B<sub>tρ</sub>pxqq `

ˆ

BρpxqzBtρpxq

apyq `b|∇_µwpyq|^pdµpyq. (25) Using the annular decay property (see Definition 4), we have

ˆ

BρpxqzBtρpxq

|∇_µw|^pdµ“ ˆ

BρpxqzBτ ρpxq

|∇_µw|^pdµ` ˆ

Bτ ρpxqzBtρpxq

|∇_µw|^pdµ

ďK₀p1´τq^ηµpB_ρpxqq|∇_µupxq|^p` ˆ

Bτ ρpxqzBtρpxq

|∇_µw|^pdµ. (26) Since ∇_µw “ ϕ∇_µv ` p1´ϕq∇<sub>µ</sub>upxq `D_µϕb pv´u_xq µ-a.e. in Ω, for some C_p ą 1 depending onponly, we have

ˆ

Bτ ρpxqzBtρpxq

|∇_µw|^pdµďC_p ˆ

Bτ ρpxqzBtρpxq

|∇_µu|^pdµ`C_pµpB_{τ ρ}pxq zB_tρpxqq|∇_µupxq|^p

`Cp

ˆ

Bτ ρpxqzBtρpxq

}Dµϕ}^p_L8

µ |u´ux|^pdµ ďC_p

ˆ

BρpxqzBtρpxq

|∇_µu|^pdµ`C_pK₀p1´tq^ηµpB_ρpxqq|∇_µupxq|^p

` C_pC₀^p pτ´tq^p

1 ρ^p

ˆ

Bρpxq

|u´u_x|^pdµ. (27)

12

Collecting (26) and (27) and dividing byµpB_ρpxqq, we have 1

µpB_ρpxqq ˆ

BρpxqzBtρpxq

|∇_µw|^pdµ

ďK₀p1´τq^η|∇_µupxq|^p` Cp

µpB_ρpxqq ˆ

BρpxqzBtρpxq

|∇_µu|^pdµ`C_pK₀p1´tq^η|∇_µupxq|^p

` CpC₀^p pτ ´tq^p

1

ρ^p _Bρpxq|u´ux|^pdµ.

Passing to the limitρÑ0, by taking (21), (19) and Lemma 2 (12) into account, we obtain limρÑ0

1 µpB_ρpxqq

ˆ

BρpxqzBtρpxq

|∇_µw|^pdµďK₀p1´τq^η|∇_µupxq|^p`2C_pK₀p1´tq^η|∇_µupxq|^p ď4C_pK₀p1´tq^η|∇_µupxq|^p. (28) Now, dividing byµpB_ρpxqqthe inequality (25) and using (22)

mpu_x, B_ρpxqq

µpB_ρpxqq ďF pv, B_tρpxqq

µpB_tρpxqq ` 1 µpB_ρpxqq

ˆ

BρpxqzBtρpxq

a`b|∇µw|<sup>p</sup>dµ ďε` mpu, B_tρpxqq

µpBtρpxqq ` 1 µpBρpxqq

ˆ

BρpxqzBtρpxq

adµ

`b 1

µpB_ρpxqq ˆ

BρpxqzBtρpxq

|∇_µw|^pdµ,

lettingρÑ0, by using Lemma 2 (12) together with (18) and (28), we have

ρÑ0lim

mpu_x, B_ρpxqq

µpB_ρpxqq ďε`lim

ρÑ0

mpu, B_tρpxqq

µpB_tρpxqq `K<sub>0</sub>p1´tq<sup>η</sup>apxq `4C_pK₀p1´tq^η|∇_µupxq|^p. (29) Since (18), (19) and the growth condition (C₄) we have

limρÑ0

mpu, B_tρpxqq µpB_tρpxqq “lim

rÑ0

mpu, B_rpxqq µpB_rpxqq ă 8.

Lettingt Ñ1in (29) we obtain limρÑ0

mpu_x, B_ρpxqq

µpB_ρpxqq ďε`lim

rÑ0

mpu, B_rpxqq µpB_rpxqq . Step 2.2: we prove that lim_ρÑ0 ^mpu_µpB^x,Bρpxqq

ρpxqq ě lim_rÑ0 ^mpu,B_µpB ^rpxqq

rpxqq for µ-a.e. x P Ω. Let ε ą 0, ρą0andtPs1,2r. There existsv P ux`H_µ,0^1,ppBρpxq;R^mqsuch that

F pv, B_ρpxqq ďεµpB_ρpxqq `mpu<sub>x</sub>, B<sub>ρ</sub>pxqq. (30) Letτ Ps1, tr. We consider a Lipschitz functionϕ: ΩÑ r0,1swhich is a Urysohn function for the pair`

ΩzB_{τ ρ}pxq, B_ρpxq˘

, i.e. satisfyingϕ”1onB_ρpxq,ϕ”0onΩzB_{τ ρ}pxqand }D_µϕ}_L8

µ ď C₀

ρpt´τq,

for some C₀ ą 0 not depending on ρ, τ and t. We set w :“ ϕv ` p1´ϕqu P u ` H_µ,0^1,ppB_ρpxq;R^mqgiven by

w“

$

&

%

v inB_ρpxq

ϕu_x` p1´ϕqu inB_{τ ρ}pxq zB_ρpxq u inB_tρpxq zB_{τ ρ}pxq.

13

Since the locality hypothesis (C₂) and (24)

mpu, B_tρpxqq ď Fpw, B_tρpxqq “F pw, B_ρpxqq `F<sup>˚</sup>pw, B<sub>tρ</sub>pxq zB<sub>ρ</sub>pxqq ďF pv, B<sub>tρ</sub>pxqq `

ˆ

BtρpxqzBρpxq

a`b|∇_µw|^pdµ. (31) Using the annular decay property, we have for someCp ą1depending onponly

ˆ

BtρpxqzBρpxq

|∇_µw|^pdµ“ ˆ

BtρpxqzBτ ρpxq

|∇_µw|^pdµ` ˆ

Bτ ρpxqzBρpxq

|∇_µw|^pdµ

ďCp

ˆ

BtρpxqzBρpxq

|∇µu|^pdµ`CpK0

ˆ 1´ 1

τ

˙η

µpBτ ρpxqq|∇µupxq|^p

` C_pC₀^pt^p pt´τq^p

1 ptρq^p

ˆ

Btρpxq

|u´u_x|^pdµ (32) dividing by µpB_tρpxqq and passing to the limitρ Ñ 0, by taking (21), (19) and (12) into account, we have

ρÑ0lim 1 µpBtρpxqq

ˆ

BtρpxqzBρpxq

|∇_µw|^pdµď2C_pK₀ ˆ

1´ 1 t

˙η

|∇_µupxq|^p. (33) Now, dividing byµpB_tρpxqqthe inequality (31) and using (30)

mpu, B_tρpxqq

µpB_tρpxqq ďε`mpu_x, B_ρpxqq

µpB_ρpxqq ` 1 µpB_tρpxqq

ˆ

BtρpxqzBρpxq

adµ

`b 1

µpB_tρpxqq ˆ

BtρpxqzBρpxq

|∇_µw|^pdµ

lettingρÑ0, by using Lemma 2 (12) together with (18) and (33), we have lim

ρÑ0

mpu, B_tρpxqq

µpB_tρpxqq ďε`lim

ρÑ0

mpu_x, B_ρpxqq µpB_ρpxqq `K₀

ˆ 1´ 1

t

˙η

papxq `2C_p|∇_µupxq|^pq. (34) Since (18), (19) and the growth condition (C₄) we have

lim

ρÑ0

mpu, B_tρpxqq µpB_tρpxqq “lim

rÑ0

mpu, B_rpxqq µpB_rpxqq ă 8.

Lettingt Ñ1in (34) we obtain limrÑ0

mpu, B_rpxqq

µpB_rpxqq ďε`lim

ρÑ0

mpu_x, B_ρpxqq µpB_ρpxqq . Step 3: proof of (i) and (ii). From Steps 1 and 2, we have

F pu, Oq “ ˆ

O

limρÑ0

mpu_x, B_ρpxqq µpB_ρpxqq dµpxq

for allpu, Oq PH_µ^1,ppΩ;R^mq ˆOpΩq. Using the extension (23) ofF pu,¨qto all Borel sets, we have

F^˚pu, Bq “ ˆ

B

limρÑ0

mpu_x, B_ρpxqq µpB_ρpxqq dµpxq

14