On the relaxation of some classes of pointwise gradient constrained energies

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On the relaxation of some classes of pointwise gradient constrained energies

Sur la relaxation de quelques classes d’énergies avec contraintes ponctuelles sur le gradient

Riccardo De Arcangelis

Università di Napoli Federico II, Dipartimento di Matematica e Applicazioni Renato Caccioppoli, via Cintia, Complesso Monte S. Angelo, 80126 Napoli, Italy

Received 14 October 2005; accepted 15 December 2005 Available online 12 July 2006

Abstract

The integral representation problem on BV(Ω) for theL¹(Ω)-lower semicontinuous envelopeFof the functional F:u∈ W^1,∞(Ω)→

Ωf (∇u)dxis approached whenf is a Borel function, not necessarily convex, with values in[0,+∞]. The pres- ence of the value+∞in the image off involves a pointwise gradient constraint on the admissible configurations, since those generating the relaxation process must satisfy the condition∇u(x)∈domf for a.e.x∈Ω. The main novelty relies in the absence of any convexity assumption on the domain off. For every convex bounded open setΩ,Fis represented on the wholeBV(Ω) as an integral of the calculus of variations by means of the convex lower semicontinuous envelope off. Due to the lack of the convexity properties of domf, the classical integral representation techniques, based on measure theoretic arguments, seem not to work properly, thus an alternative approach is proposed. Applications are given to the relaxation of Dirichlet variational problems and to first order differential inclusions.

©2006 Elsevier Masson SAS. All rights reserved.

Résumé

Le problème de représentation intégrale surBV(Ω)pour l’enveloppeL¹(Ω)-semi-continue inférieurementFde la fonction- nelleF:u∈W^1,∞(Ω)→

Ωf (∇u)dxest considéré dans le cas oùfest Borelienne non nécessairement convexe, à valeurs dans [0,+∞]. La présence de la valeur+∞dans l’image def implique une contrainte ponctuelle sur le gradient des configurations admissibles, puisque celles qui jouent un rôle dans le processus de relaxation doivent satisfaire la condition∇u(x)∈domf p.p.

dansΩ. La nouveauté principale consiste en l’absence d’hypothèses de convexité sur le domaine def. Pour tout ensemble ouvert borné et convexeΩ,Fadmet une représentation surBV(Ω)tout entier comme une intégrale du calcul des variations au moyen de l’enveloppe convexe et semi-continue inférieurement def. En raison du manque des propriétés de convexité de domf, les techniques classiques de représentation intégrale, basées sur des arguments de théorie de la mesure, semblent ne pas fonction- ner convenablement, donc une approche alternative est proposée. Des applications sont données à la relaxation des problèmes variationnels de type Dirichlet et aux inclusions différentielles du premier ordre.

©2006 Elsevier Masson SAS. All rights reserved.

E-mail address:[email protected] (R. De Arcangelis).

0294-1449/$ – see front matter ©2006 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2005.12.003

MSC:49J45; 49J24; 49J40

Keywords:Relaxation; Pointwise gradient constraints; Nonconvex variational problems;BVspaces; First order differential inclusions

0. Introduction

LetUbe a set, and letF:U→ [0,+∞]. Then the approach to the minimization problem forF onUby means of the direct methods of the calculus of variations naturally leads to the introduction of a topologyτ onUenjoying good compactness properties, and to the study of theτ-lower semicontinuity ofF. IfF is notτ-lower semicontinuous, one is naturally led to introduce theτ-relaxed functionalFofF, defined on theτ-closureUofU as the greatestτ-lower semicontinuous functional onUless than or equal toF onU. Indeed,Fturns out to beτ-lower semicontinuous, the minimum ofFonUexists providedF satisfies suitable coerciveness conditions, and

min

u∈U

F (u) = inf

u∈UF (u).

Whenf:Rⁿ→ [0,+∞[is Borel,Ωis a smooth bounded open subset ofRⁿ,U⊆W^1.1(Ω), andF is the integral energy defined by (here and in the followingLⁿdenotes the Lebesgue measure onRⁿ)

F:u∈U→

Ω

f (∇u)dLⁿ, (0.1)

the above described relaxation process has been widely developed in the last decades for various choices ofU, each one determining a particular variational problem (Neumann, Dirichlet, mixed, etc.), and under different assumptions onf. We refer to the books [1,2,8,11,14,20,31] for complete references on the subject, also in more general frame- works.

By choosingτ equal to theL¹(Ω)-topology, and under convexity assumptions onf, in [23] and in [10] the case of the Neumann problem has been treated whenU is a Sobolev space, or a space of smooth functions. In these papers, integral representation results forFhave been proved on the spaceBV(Ω)of the functions with bounded variation inΩ. In the same framework, in [22] and [16] the case of the Dirichlet problem has been treated by imposing a boundary trace condition on the elements ofU, and again proving integral representation results forFonBV(Ω).

Whenf is not convex, relaxation processes have been carried out in both the cases of Neumann and Dirichlet problems whenUis a Sobolev space, or a space of smooth functions, andτ is either the sequential weak-W^1,p(Ω) topology, withp depending on the choice ofU, or the L¹(Ω) one (cf. for example [2,9,14,20,30] and the refer- ences quoted therein). In these papers integral representation results forFhave been proved in Sobolev spaces also in more general settings, for example by allowing a dependence off also on the space variablex, under additional coerciveness and growth assumptions. It has been shown that the relaxation process produces a density convexifi- cation. For example, whenU=W^1,∞(Ω)andτ agrees with the sequential weak*-W^1,∞(Ω)-topology, it turns out that

F (u) =

Ω

f^∗∗(∇u)dLⁿ for everyu∈W^1,∞(Ω), (0.2)

wheref^∗∗is the convex envelope off, i.e. the greatest convex function less than or equal tof.

We point out that results in the same spirit hold also in different settings. For example, whenf is defined on the set of then×mmatrices and the elements ofU areR^m-valued, (0.2) still holds providedf^∗∗ is replaced by the quasiconvex envelope off (cf. for example [2,11,31]).

In the above results the gradients of the elements ofUare allowed to lie in the whole ofRⁿwithout any restriction.

When this does not occur, namely when a condition like (unless differently specified, a.e. meansLⁿ-a.e.)

∇u(x)∈E for a.e.x∈Ω,

must be fulfilled by the elements ofUfor some given subsetEofRⁿ, the corresponding relaxation processes become pointwise gradient constrained. The treatment of this case can be handled by allowing the value+∞in the target

space off. Indeed, in this case the only elements ofU that play a role are those that satisfy the following pointwise gradient constraint

∇u(x)∈domf for a.e.x∈Ω, (0.3)

where domf = {z∈Rⁿ: f (z) <+∞}.

Several situations in applications, for example in elastic-plastic torsion theory, in nonlinear elastomers modeling, and in optimal control problems, lead to classes of variational inequalities and of relaxation problems on sets of admissible configurations subject to pointwise gradient constraints of the above type (cf. for example [8,19,27,28,34, 35] and the references quoted therein).

It is clear that, in general, (0.3) can be a very restrictive condition, entailing serious technical difficulties and hindering the development of a wide range of results like those described in the unconstrained case. Indeed, few results exist in literature on pointwise gradient constrained relaxation. We quote [20,30,5–7], and the monograph [8] in which additional gradient constrained variational problems are considered as well. In particular, we quote [4,18], and [3] for the treatment of the corresponding homogenization processes. In these papers, under various sets of assumptions onf, and with different choices ofU andτ, again formulas similar to (0.2) have been proved, where now, sincef takes its values in[0,+∞],f^∗∗is the convex lower semicontinuous envelope off. In particular, in [5] and [6] these formulas have been extended toBV spaces as well, and some cases in which domf has empty interior have been treated. In spite of this, it must be emphasized that all these papers assume the structure condition

domf is convex, (0.4)

that however forestalls the approach in this context to the cases in which the gradients of the admissible configurations lie in disconnected or finite sets.

Finally, we point out that recently, in [15] and [17], gradient constrained relaxation processes for Neumann prob- lems have been investigated by allowing a true dependence on the space variable x inf, either under continuity or just measurability assumptions on the multifunction

x→domf (x,·), (0.5)

but assuming that for a.e.x, domf (x,·)is convex, uniformly bounded, and with nonempty interior. In these papers a formula like

F (u) =

Ω

f^∗∗(x,∇u)dLⁿ for everyu∈W^1,∞(Ω), (0.6)

where for a.e.x,f^∗∗(x,·)is the convex lower semicontinuous envelope off (x,·), has been proved under continuity type assumptions on the multifunction in (0.5), but has been shown to fail under just measurability ones. Nevertheless, in this last case, it must be pointed out thatFstill has an integral form as in (0.6), but withf^∗∗replaced by a suitable integrandf¯= ¯f (x, z)convex and lower semicontinuous in thezvariable.

In the present paper we study pointwise gradient constrained relaxation processes for functionals as in (0.1) when assumption (0.4) is dropped.

Actually, very little is known on this problem, and the measure theoretic techniques developed in the above men- tioned papers seem not to be well suited for this case. Consequently, we propose an approach based on a new technique, that allows us to treat both the cases of Neumann and Dirichlet problems. More precisely, iff:Rⁿ→ [0,+∞]is Borel andΩis a convex bounded open subset ofRⁿ, we prove, in the case of Neumann problems withU=W^1,∞(Ω)and τ equal to theL¹(Ω)-topology, that (cf. Theorem 3.10)

F (u) =

Ω

f^∗∗(∇u)dLⁿ+

Ω

f^∗∗_∞ dD^su d|D^su|

dD^su for everyu∈BV(Ω),

where(f^∗∗)^∞ is the recession function off^∗∗ (cf. Section 1 for the definition), and, for everyu∈BV(Ω),∇u is the density of theLⁿ-absolutely continuous part of theRⁿ-valued measure gradient ofu,D^suis theLⁿ-singular part of the gradient ofu,|D^su| is its total variation, and _d^dD_|D^ss^uu| is the Radon–Nikodym derivative ofD^su with respect to|D^su|.

Of course, the above formula agrees with the above recalled one established under (0.4), but we emphasize that here we do not need to assume any topological or geometrical condition on domf.

We also observe explicitly that the constraint condition involved in the relaxed problem, at least on Sobolev func- tions, is given by

∇u(x)∈co(domf ) for a.e.x∈Ω, where co(domf )is the convex envelope of domf.

In the case of Dirichlet problems, we first remark that the only nontrivial results occur when co(domf ) has nonempty interior (cf. Proposition 4.1). Then we take z₀∈int(co(domf )), and consider the case in which U= u_z0 +W₀^1,∞(Ω) and τ is again the L¹(Ω)-topology, where u_z0 is the linear function whose gradient is z₀ and W₀^1,∞(Ω)the set of the Lipschitz functions onΩ whose (unique) extension onΩ is equal to 0 on∂Ω. We prove that (cf. Theorem 4.5)

F (u) =

Ω

f^∗∗(∇u)dLⁿ+

Ω

f^∗∗_∞ dD^su d|D^su|

dD^su+

∂Ω

f^∗∗_∞

(u_z0−u)nΩ

dHⁿ⁻¹

for everyu∈BV(Ω),

wherenΩ is the unit outward normal to∂Ω andHⁿ⁻¹the(n−1)-dimensional Hausdorff measure.

To prove the above results, in both the cases forU, we proceed by means of successive representations ofFon wider and wider function classes, starting from the one of affine configurations. The main novelty of the paper is just in the techniques introduced to representFon such space and on the one of piecewise affine functions. By improving an idea from [24], we are able to approximate every linear functionu_zby means of a sequence of functions{u_h}whose gradients take only a finite number of values in domf and such that limh→+∞

Ωf (∇u_h)dLⁿf^∗∗(z)Lⁿ(Ω)(cf.

Proposition 3.5). A refinement of this result also provides a tool to replace the boundary datum of a function with a prescribed linear one without perturbing too much the corresponding energy, thus allowing the treatment of Dirichlet problems. It also generalizes the so called zig–zag lemma (cf. [14]) to the case of convex combinations of more than two vectors. The representation ofFon piecewise affine functions is then deduced by means of a structure result establishing that every piecewise affine functionu= _j(u_zj +s_j)χ_Pj can be expressed at each point of a convex open setΩ as a maximum of minima of some of its componentsu_zj +s_j whose correspondingP_j have a nonempty intersection withΩ (cf. Theorem 2.1). Finally, the representation onBV spaces is achieved by means of suitable approximation processes, and of a general inner regularity result for abstract functionals (cf. Proposition 1.7).

We point out that pointwise gradient constrained relaxation problems are related to first order differential inclusions and Hamilton–Jacobi equations (cf. [29,13], and also [12] where existence results of a.e. solutions of differential inclusions are proved without assuming convexity hypotheses on the inclusion sets). In fact (for simplicity we discuss only the particular case of Sobolev functions), when applied tof =IE, whereE is a Borel subset ofRⁿ andIE is its indicator function defined asI_E(z)=0 ifz∈EandI_E(z)= +∞ifz∈Rⁿ\E, our results imply that for every u∈W^1,1(Ω)satisfying∇u(x)∈co(E)for a.e.x∈Ω, there exists{u_h}inW^1,∞(Ω)such thatu_h→uinL¹(Ω), and∇u_h(x)∈Efor everyh∈Nand a.e.x∈Ω(cf. Corollary 3.12). In addition, if int(co(E))= ∅,z₀∈int(co(E)), and theuabove is inu_z0+W₀^1,1(Ω), then{u_h}can be taken inu_z0+W₀^1,∞(Ω)(cf. Corollary 4.7).

In both these results{u_h}can be any sequence of solutions of the differential inclusion∇v∈Ea.e. inΩ, possi- bly satisfying a boundary condition. Conversely, we emphasize that, whenf is not merely an indicator function, if u∈W^1,1(Ω) satisfies

Ωf^∗∗(∇u)dLⁿ <+∞, an additional difficulty occurs in the selection of the optimal sequences {uh}, converging to u in L¹(Ω), provided by Theorem 3.10. Indeed, beside the differential inclusion

∇u_h(x)∈domf for every h∈Nand a.e. x ∈Ω, they must satisfy also the additional minimality condition ex- pressed by the convergence of{

Ωf (∇u_h)dLⁿ}to

Ωf^∗∗(∇u)dLⁿ. Analogous remarks hold in the case of Dirichlet problems and Theorem 4.5 as well.

Eventually, we observe that our results are connected to those of the recent papers [25] and [26], where a relaxation phenomenon for Hamilton–Jacobi equations is pointed out by showing that the pointwise supremum of certain a.e.

subsolutions of a Hamilton–Jacobi equation yields a viscosity solution of the corresponding convexified equation.

1. Recalls and preliminary results

In the present section we recall some notions, and prove some preliminary result, needed in the paper. Eventually, we establish some notations on the relaxed functionals we will be concerned with.

1.1. Recalls of convex analysis

We recall here some basics of convex analysis. We refer to [32] and [33] for a more complete exposition of the matter.

For a given S⊆Rⁿ we denote by aff(S) the affine hull of S, defined as the intersection of all the affine sets containingS. It is clear that aff(S)is the smallest affine set containingS.

For everyS⊆Rⁿwe denote by co(S)the convex hull ofS, i.e. the intersection of all the convex subsets ofRⁿ containingS. It is clear that co(S)is the smallest convex set containingS.

If C⊆Rⁿ is convex, we denote by ri(C) the relative interior ofC, i.e. the set of the interior points ofC, in the topology of aff(C), once we regard it as a subspace of aff(C). We recall that ri(C)= ∅providedC= ∅. When aff(C)=Rⁿwe write as usual ri(C)=int(C). Moreover, we also recall that

t z+(1−t )z0∈ri(C) wheneverz0∈ri(C), z∈ C, andt∈ [0,1[. (1.1) For ν∈ {0, . . . , n} and z₀, . . . , z_ν ∈ Rⁿ, we say that z₀, . . . , z_ν are affinely independent if the dimension of aff({z₀, . . . , z_ν})is ν. We recall that ifz₀, . . . , z_ν are affinely independent, then the expression of each element of co({z0, . . . , zν})as a convex combination ofz0, . . . , zν is unique. In addition, ifν=n, then int(co({z0, . . . , zn}))= ∅ and

int co

{z₀, . . . , z_n}

= _n

j=0

t_jz_j: t_j∈ ]0,1[for everyj ∈ {0, . . . , n}, n j=0

t_j=1

. (1.2)

A subset P of Rⁿ is said to be a polyhedral set if it is the intersection of a finite family of closed half-spaces.

Clearly, a polyhedral set is closed and convex. Moreover, a bounded polyhedral set turns out to be the convex envelope of finitely many of its points.

For everyC⊆Rⁿwith 0∈C, the polarC^◦ofCis defined by C^◦=

x∈Rⁿ:z·x1 for everyz∈C .

It is clear that polar sets are closed and convex. The result below describes some of their additional properties (cf.

11.20 Exercise in [33]).

Proposition 1.1.LetC⊆Rⁿbe closed, convex, and with0∈C. Then Cis bounded if and only if0∈int

C^◦ , C^◦is bounded if and only if0∈int(C), and

Cis a polyhedral set if and only if so isC^◦.

Eventually, we recall that for everyC⊆Rⁿ, the support functionσCofCis defined by σ_C:x∈Rⁿ→sup{z·x: z∈C}.

It is clear that support functions are convex, lower semicontinuous, and positively 1-homogeneous. Moreover, ifC⊆Rⁿsatisfies 0∈C, it is easy to verify that

0σ_C(x)1 for everyx∈C^◦, (1.3)

and, by using Proposition 1.1, that

σ_C(x)=1 for everyx∈∂C^◦, providedCis compact and convex. (1.4)

Letf:Rⁿ→ [0,+∞]be convex. Then it is well known that domf is convex, that f is lower semicontinuous in ri(domf ), and that the restriction off to ri(domf ) is continuous. In particular, if int(domf )= ∅, then f is continuous in int(domf ).

For everyf:Rⁿ→ [0,+∞]we denote by cof the convex envelope off, i.e. the function cof:z∈Rⁿ→sup

φ(z): φ:Rⁿ→ [0,+∞]convex, φ(ζ )f (ζ )for everyζ∈Rⁿ .

Clearly, cof is convex, and cof (z)f (z)for everyz∈Rⁿ. Consequently, cof turns out to be the greatest convex function on Rⁿ less than or equal tof. The representation result below comes from Carathéodory Theorem (cf.

Corollary 17.1.3 in [32]).

Theorem 1.2.Letf:Rⁿ→ [0,+∞]. Then, for everyz∈Rⁿ, cof (z)=inf

ν j=0

t_jf (z_j),

where the infimum is taken over all theν∈ {0, . . . , n},z0, . . . , zν∈Rⁿ, andt0, . . . , tν∈ ]0,1]such thatz0, . . . , zν are affinely independent, ^ν_j=0t_j=1, and ^ν_j=0t_jz_j=z.

For everyf:Rⁿ→ [0,+∞]we denote byf^∗∗the convex lower semicontinuous envelope off, i.e. the function defined by

f^∗∗:z∈Rⁿ→sup

φ(z): φ:Rⁿ→ [0,+∞]convex and lower semicontinuous, φ(ζ )f (ζ ) for everyζ ∈Rⁿ

.

Clearly,f^∗∗is convex and lower semicontinuous, andf^∗∗(z)f (z)for everyz∈Rⁿ. Consequently,f^∗∗turns out to be the greatest convex lower semicontinuous function onRⁿless than or equal tof.

Proposition 1.3.Letf:Rⁿ→ [0,+∞]. Thenri(domf^∗∗)=ri(dom(cof ))=ri(co(domf ), and f^∗∗(z)=cof (z) for everyz∈ri

co(domf )

∪

Rⁿ\co(domf ) .

We now define recession functions. To do it properly, we recall that for a givenf:Rⁿ→ [0,+∞]convex, and z0∈domf, the limit limt→+∞(f (z0+t z)−f (z0))/t exists for everyz∈Rⁿ. Therefore we define the recession function off by

f^∞:z∈Rⁿ→ lim

t→+∞

f (z₀+t z)−f (z₀)

t .

It is well known thatf^∞is positively 1-homogeneous, and that, if in additionf is also lower semicontinuous, then the definition off^∞does not depend onz₀when it varies in domf.

1.2. Lower semicontinuity results in BV spaces

LetΩ be an open subset of Rⁿ. By BV(Ω)we denote the set of the functions inL¹(Ω)having distributional partial derivatives that are Borel measures with bounded total variation inΩ.BV(Ω)is a Banach space with the norm u∈BV(Ω)→ uL¹(Ω)+ |Du|(Ω), where, for everyu∈BV(Ω),Dudenotes theRⁿ-valued measure gradient ofu, and|Du|its total variation. We refer, for example, to [36] for a complete treatment ofBVspaces.

IfΩ has Lipschitz boundary, then functions inBV(Ω)have traces on∂Ω in the sense that for everyu∈BV(Ω) there exists an element inL¹(∂Ω), still denoted byu, such that

Ω

udivϕdLⁿ= −

Ω

ϕ·dDu+

∂Ω

ϕ·nΩudHⁿ⁻¹ for everyϕ∈ C¹

Rⁿn

.

We also recall that, ifΩis another open subset ofRⁿsuch thatΩ⊆Ω, andv∈BV(Ω\Ω), then the functionw, defined a.e. inΩbyw=uinΩandw=vinΩ\Ω, is inBV(Ω), and

Dw(E)=

E

(v−u)nΩdHⁿ⁻¹ for every Borel setE⊆∂Ω. (1.5)

ByBVloc(Rⁿ)we denote the set of the functions inL¹_loc(Rⁿ)that are inBV(Ω)for every bounded open setΩ. We recall thatBVloc(Rⁿ)is a Fréchet space.

ByA0(Rⁿ)we denote the set of the bounded open subsets ofRⁿ.

Iff:Rⁿ→ [0,+∞]is convex and lower semicontinuous, we define the functionalF_f as F_f:(Ω, u)∈A0

Rⁿ

×BVloc

Rⁿ

→

Ω

f (∇u)dLⁿ+

Ω

f^∞ dD^su

d|D^su|

dD^su. (1.6)

It is clear that F_f(Ω, u)=

Ω

f (∇u)dLⁿ for every(Ω, u)∈A0

Rⁿ

×W_loc^1,1 Rⁿ

. (1.7)

The following lower semicontinuity property holds (cf. for example Theorem 7.4.6 in [8]).

Proposition 1.4.Letf:Rⁿ→ [0,+∞]be convex and lower semicontinuous, and letF_f be defined in(1.6). Then, for everyΩ∈A0(Rⁿ),F_f(Ω,·)isL¹_loc(Ω)-lower semicontinuous.

Analogously, iff is a s above,z0∈domf, andΩ∈A0(Rⁿ)has Lipschitz boundary, we defineFf(uz₀, Ω,·)as F_f(u_z0, Ω,·):u∈BVloc

Rⁿ

→

Ω

f (∇u)dLⁿ+

Ω

f^∞ dD^su

d|D^su|

dD^su+

∂Ω

f^∞

(u_z0−u)nΩ

dHⁿ⁻¹. (1.8)

It is clear that F_f(u_z0Ω, u)=

Ω

f (∇u)dLⁿ for everyΩ∈A0

Rⁿ

with Lipschitz boundary, u∈u_z0+W₀^1,1(Ω). (1.9) Since (1.5), and the 1-homogeneity off^∞imply

F_f(u_z0Ω, u)=F_f(Ω, u)−f (z₀)Lⁿ(Ω\Ω) wheneverz₀∈domf, Ω∈A0

Rⁿ

has Lipschitz boundary, Ω∈A0

Rⁿ

satisfiesΩ⊆Ω, andu∈BV(Ω)is such thatu=u_z0 a.e. inΩ\Ω, the lower semicontinuity result below follows from Proposition 1.4.

Proposition 1.5.Letf:Rⁿ→ [0,+∞]be convex and lower semicontinuous, and letz₀∈domf. LetΩ ∈A0(Rⁿ) have Lipschitz boundary, and letF_f(u_z0, Ω,·)be defined in(1.8). ThenF_f(u_z0, Ω,·)isL¹(Ω)-lower semicontinu- ous.

LetB₁be the unit open ball ofRⁿcentred in 0, and letαbe a symmetric mollifier, namely a nonnegative function inC₀^∞(B₁), symmetric with respect to 0, and such that

B1αdLⁿ=1. For everyu∈L¹_loc(Rⁿ),η >0, andx∈Rⁿ, let us denote byu_η(x)the regularization ofuatxdefined by

u_η(x)= 1 ηⁿ

Rⁿ

α x−y

η

u(y)dy.

It is well known that for everyη >0,u_η∈C^∞(Rⁿ), and thatu_η→uinL¹_loc(Rⁿ)asη→0.

The following approximation in energy result forF_f holds (cf. Lemma 7.4.4 in [8]). In it and in the following, for everyΩ∈A0(Rⁿ)andη >0, we setΩ_η⁻= {x∈Ω: dist(x, ∂Ω) > η}.

Proposition 1.6.Letf:Rⁿ→ [0,+∞]be convex and lower semicontinuous, and letFf be defined in(1.6). Then F_f

Ω_η⁻, u_η

F_f(Ω, u) for everyΩ∈A0

Rⁿ

, u∈BVloc

Rⁿ

,andη >0.

1.3. Measure theoretic preliminaries

Letϑ:A0(Rⁿ)→ [0,+∞]. We say thatϑis increasing if ϑ (Ω₁)ϑ (Ω₂) for everyΩ₁, Ω₂∈A0

Rⁿ

such thatΩ₁⊆Ω₂. We denote byϑ₋the inner regular envelope ofϑdefined by

ϑ₋:Ω∈A0

Rⁿ

→sup

ϑ (A): A∈A0

Rⁿ

, A⊆Ω , and say thatϑis inner regular inΩ∈A0(Rⁿ)if

ϑ (Ω)=ϑ₋(Ω).

It is clear thatϑ₋is increasing, and that it is inner regular inΩ for everyΩ∈A0(Rⁿ).

Hereafter, ifUis a set,Φ:A0(Rⁿ)×U→ [0,+∞], andu∈U, we denote byΦ₋(·, u)the inner regular envelope ofΦ(·, u), namely the function defined, for everyΩ∈A0(Rⁿ), byΦ₋(Ω, u)=Φ(·, u)₋(Ω).

For everyS⊆Rⁿ, every functionuonS,x0∈Rⁿ, andt >0, we defineux₀,tas u_x0,t:x∈x₀+1

t(S−x₀)→1 tu

x₀+t (x−x₀)

. (1.10)

The following inner regularity criterion is proved, also in a more general setting, in Proposition 2.7.4 of [8].

Proposition 1.7.LetΦ:A0(Rⁿ)×BVloc(Rⁿ)→ [0,+∞]be such that for everyu∈BVloc(Rⁿ),Φ(·, u)is increasing.

LetΩ∈A0(Rⁿ)be convex,u∈BVloc(Rⁿ),x₀∈Ω, and assume that lim inf

t→1⁻

Φ(Ω, u_x0,t)Φ(Ω, u), lim sup

t→1+ Φ₋

x0+t (Ω−x0), ux₀,1/t

Φ₋(Ω, u).

Then

Φ(Ω, u)=Φ₋(Ω, u).

The following paving result comes from the Vitali Covering Theorem.

Proposition 1.8.LetΩ⊆Rⁿbe open, and letKbe a compact subset ofRⁿ such that0∈K andLⁿ(K) >0. Then there exist{x_h} ⊆Ωand{t_h} ⊆ ]0,1]such that the sets{x_h+t_hK: h∈N}are contained inΩ, pairwise disjoint, and

Lⁿ

Ω

h∈N

(x_h+t_hK)

=0.

Proof. The Vitali Covering Theorem (cf. for example Corollary 2 at page 28 of [21]) provides a sequence{C_h}of pairwise disjoint closed balls inΩwith radius less than or equal to diam(K), andLⁿ(Ω\

h∈NC_h)=0. Let{B_h}the sequence of the corresponding open balls, then it is easy to verify that alsoLⁿ(Ω\

h∈NB_h)=0. LetBbe an open ball centred in 0 and with radius strictly greater than diam(K). Then, if for everyh∈N,x_h¹is the centre ofB_handr_h¹ is its radius divided by the radius ofB, we deduce that{r_h¹} ⊆ ]0,1], and, since 0∈K, thatx_h¹+r_h¹K⊆x_h¹+r_h¹B=B_h for everyh∈N. Because of this, the sets{x_h¹+r_h¹K: h∈N}too are pairwise disjoint, and

Lⁿ

Ω

h∈N

x_h¹+r_h¹K

=Lⁿ

h∈N

B_h

h∈N

x_h¹+r_h¹K

=

h∈N

Lⁿ

x_h¹+r_h¹B

x_h¹+r_h¹K

=Lⁿ(B\K) Lⁿ(B)

h∈N

r_h¹n

Lⁿ(B)=Lⁿ(B\K) Lⁿ(B) Lⁿ(Ω).

Now, let us setΩ1=(

h∈NBh)\

h∈N(x¹_h+r_h¹K). Then, sinceΩ1=

h∈N(Bh\(x_h¹+r_h¹K)),Ω1turns out to be open. Consequently, we can repeat the above argument starting fromΩ₁in place ofΩ, thus getting{x_h²} ⊆Ω₁and {r_h²} ⊆ ]0,1]such that the sets{xⁱ_h+r_hⁱK: i∈ {1,2}, h∈N}are pairwise disjoint,

Lⁿ

Ω₁

h∈N

x_h²+r_h²K

=Lⁿ(B\K)

Lⁿ(B) Lⁿ(Ω₁)=Lⁿ(B\K) Lⁿ(B) Lⁿ

h∈N

B_h

h∈N

x_h¹+r_h¹K

=Lⁿ(B\K) Lⁿ(B) Lⁿ

Ω

h∈N

x_h¹+r_h¹K

=

Lⁿ(B\K) Lⁿ(B)

2

Lⁿ(Ω),

and Lⁿ

Ω

i∈{1,2}

h∈N

x_hⁱ +r_hⁱK

=Lⁿ

Ω1 h∈N

x_h²+r_h²K

=

Lⁿ(B\K) Lⁿ(B)

2

Lⁿ(Ω).

By iterating the above argument, we obtain that for everym∈Nthere exist{x_h^m} ⊆Ω and{r_h^m} ⊆ ]0,1]such that the sets{x_hⁱ +r_hⁱK: i∈ {1, . . . , m}, h∈N}are pairwise disjoint, and

Lⁿ

Ω

i∈{1,...,m}

h∈N

x_hⁱ +r_hⁱK

=

Lⁿ(B\K) Lⁿ(B)

m

Lⁿ(Ω).

Because of this, we conclude that the sets {xⁱ_h+r_hⁱK: i∈N, h∈N}satisfy the properties required in the proposi- tion. 2

1.4. Relaxed functionals

Here we define the relaxed functionals that we will consider in this the paper.

For everyf:Rⁿ→ [0,+∞]Borel,Ω∈A0(Rⁿ), andu0∈W^1,∞(Ω), we define F (Ω, ·):u∈L¹(Ω)→min

lim inf

h→+∞

Ω

f (∇u_h)dLⁿ: {u_h} ⊆W^1,∞(Ω), u_h→uinL¹(Ω)

(1.11) and, in order to analyze Dirichlet type problems,

F (u ₀, Ω,·):u∈L¹(Ω)→min

lim inf

h→+∞

Ω

f (∇u_h)dLⁿ: {u_h} ⊆u₀+W₀^1,∞(Ω), u_h→uinL¹(Ω)

, (1.12) where the minima are trivially attained, since theL¹(Ω)-topology is metric.

For technical reasons, and to deduce sharper estimates as well, we need to introduce the two additional relaxed functionals below, analogous to the above ones, but in the uniform convergence topology. To this aim, when no confusion occurs, we denote, for everyΩ∈A0(Rⁿ), byC⁰(Ω)both the space of the continuous functions onΩand the usual topology of the uniform convergence onΩ. We define

G(Ω, ·):u∈C⁰ Ω

→min

lim inf

h→+∞

Ω

f (∇u_h)dLⁿ: {u_h} ⊆W^1,∞(Ω), u_h→uinC⁰

Ω

(1.13) and

G(u 0, Ω,·):u∈C⁰ Ω

→min

lim inf

h→+∞

Ω

f (∇uh)dLⁿ: {uh} ⊆u0+W₀^1,∞(Ω), uh→uinC⁰

Ω

. (1.14) We recall that, for everyΩ∈A0(Rⁿ)andu0∈W^1,∞(Ω),F (Ω, ·)andF (u 0, Ω,·)areL¹(Ω)-lower semicontin- uous, and thatG(Ω, ·)andG(u 0, Ω,·)areC⁰(Ω)-lower semicontinuous. Obviously,

F (Ω, u) G(Ω, u), F (u ₀, Ω, u)G(u ₀, Ω, u) for everyu∈C⁰ Ω

. (1.15)

Finally, we observe that (1.7) and Proposition 1.4 provide

Ω

f^∗∗(∇u)dLⁿ+

Ω

f^∗∗_∞ dD^su d|D^su|

dD^suF (Ω, u) for everyΩ∈A0

Rⁿ

, u∈BV(Ω), (1.16)

whilst (1.9) and Proposition 1.5 imply

Ω

f^∗∗(∇u)dLⁿ+

Ω

f^∗∗_∞ dD^su d|D^su|

dD^su+

∂Ω

f^∗∗_∞

(u_z0−u)nΩ

dHⁿ⁻¹F (u _z0, Ω, u)

for everyΩ∈A0

Rⁿ

with Lipschitz boundary, u∈BV(Ω). (1.17)

2. A representation result for piecewise affine functions

For everyE⊆Rⁿ, we denote byχ_E the characteristic function ofEdefined asχ_E(x)=1 ifx∈Eandχ_E(z)=0 ifx∈Rⁿ\E.

Letube a continuous function onRⁿ. We say thatuis piecewise affine if u(x)=

m j=1

u_zj(x)+s_j

χ_Pj(x) for a.e.x∈Rⁿ. (2.1)

for somem∈N,z₁, . . . , z_m∈Rⁿ,s₁, . . . , s_m∈R, and some polyhedral sets P₁, . . . , P_m with nonempty pairwise disjoint open interiors such that_m

j=1P_j=Rⁿ. We observe explicitly that the presence of the a.e. requirement in (2.1) is simply due to the closedness of polyhedral sets. Actually (2.1) holds for everyxin_m

j=1int(Pj).

We denote byPA(Rⁿ)the set of the piecewise affine functions.

In the theorem below we prove that every piecewise affine functionuas in (2.1) can be represented on a convex open setΩ as a maximum of minima of some of its componentsu_zj +s_j for which int(Pj)∩Ω = ∅. The result has been already proved in [4] in the framework of homogenization problems. Here we propose a more direct and elementary proof.

Givenx₁, x₂∈Rⁿ, we set[x₁, x₂] = {t x₁+(1−t )x₂: t∈ [0,1]}, and so on for]x₁, x₂],[x₁, x₂[,]x₁, x₂[.

Theorem 2.1.Letu= ^m_j=1(uz_j+sj)χP_j be in PA(Rⁿ), and letΩ be a convex open subset ofRⁿ. Then there exist k∈NandN₁, . . . , N_k⊆ {j∈ {1, . . . , m}: int(Pj)∩Ω= ∅}such that

u(x)= max

i∈{1,...,k}min

j∈Ni

u_zj(x)+s_j

for everyx∈Ω.

Proof. For simplicity, let us assume that for somem_Ω ∈ {1, . . . , m}, int(Pj)∩Ω= ∅if and only ifj∈ {1, . . . , mΩ}.

Let us denote byI the set of the indexes in{1, . . . , mΩ}corresponding to the truly different components ofuliving inΩ, namelyI = {1} ∪ {j ∈ {2, . . . , mΩ}: z_j=z_i ors_j =s_i for everyi∈ {1, . . . , j−1}}, and letAbe the subset of Ω where such components are different, i.e. A=Ω \ {x∈Ω: there existi, j ∈I withi=j anduz_i(x)+si= uz_j(x)+sj}. It is clear thatAis open, dense inΩ, and that it possesses a finite number of connected components, say A1, . . . , Ak. We observe that for everyi∈ {1, . . . , k},Ai is open, thatuturns out to be affine inAi, and that

for everyp, q∈Iwithp=q it results thatu_zp(x)+s_p> u_zq(x)+s_qfor everyx∈A_i,

or thatu_zp(x)+s_p< u_zq(x)+s_qfor everyx∈A_i. (2.2) Let us prove that for everyi∈ {1, . . . , k}there existsN_i⊆Isuch that, if

vi:x∈Rⁿ→min

j∈N_i

uz_j(x)+sj

,

then

v_i(x)=u(x) for everyx∈A_i, (2.3)

and

vi(x)u(x) for everyx∈Ω. (2.4)

To do this, leti∈ {1, . . . , k}, and let us define N_i=

j∈I: u_zj(x)+s_ju(x)for everyx∈A_i .

Then, sinceuis affine inAi, it is clear that (2.3) holds.

To prove (2.4), we take for the momentx∈A\A_i, and fixx₀∈A_i. Then (2.3) implies that{y∈ [x₀, x]: v_iu in[x₀, y]}is nonempty, so that it possesses a point that is the farthest fromx₀. Callx₁such point, and observe that, by the continuity ofv_i and ofu,

v_i(x₁)=u(x₁).

Moreover, by (2.3),x₁∈/A_i and, in particular,x₁=x₀. If

vi(x) > u(x), (2.5)

x1turns out to be different fromx, hence there existsx2∈ ]x1, x[such that v_i(x₂) > u(x₂).

Now, we observe that the convexity ofΩ implies[x₀, x] ⊆Ω, so thatx₂∈Ω. Moreover, sincex∈AandΩ\Ais made up by the intersection of a finite numbers of hyperplanes withΩ,x₂too can be taken inAand so close tox₁to belong to the same connected component ofAwhose closure containsx1. In such connected component,u=u_zl+s_l for somel∈I, thus we have thatvi(x1)=uz_l(x1)+sl andvi(x2) > uz_l(x2)+sl. Consequently, the concavity ofvi

implies thatvi< uz_l+sl in[x0, x1[, and, sincex1=x0, that vi(x0) < uz_l(x0)+sl.

Once we recall thatuis affine inA_i, (2.2), the previous inequality, and (2.3) imply thatu < u_zl+s_linA_i, and hence, by the definition ofvi, that

v_i(x₂)u_zl(x₂)+s_l=u(x₂).

This yields a contradiction, sincev_i(x₂) > u(x₂). Consequently, (2.5) cannot hold, and, by using also (2.3), we con- clude thatv_i(x)u(x)for everyx∈A. Because of this, and of the continuity ofv_i and ofu, we deduce (2.4).

By (2.3) and (2.4), the theorem follows. 2

We conclude this section by recalling that Example 2.2 in [4] shows that, in general, in Theorem 2.1 the convexity assumption onΩcannot be dropped, and not even replaced by a connectedness one.

3. Relaxation of gradient constrained Neumann problems

Letf:Rⁿ→ [0,+∞]be Borel. In this section we prove the representation result for the functionalFin (1.11).

We start with some preparatory convex analysis lemmas. Since the first two need to be established in an R^ν-space, with ν not necessarily equal ton, the symbol ∇ and the expression a.e. used there will be referred to theR^ν-environment and toL^ν respectively.

Lemma 3.1.Letν∈N, letζ₀, . . . , ζ_ν ∈R^ν be affinely independent, lett₀, . . . , t_ν∈ ]0,1[be such that ^ν_j=0t_j =1 and ^ν_j=0t_jζ_j =0, and let ω∈A0(R^ν). Then there existsv∈W₀^1,∞(ω) such that−1v(x) <0 and ∇v(x)∈ {ζ₀, . . . , ζ_ν}for a.e.x∈ω, andL^ν({x∈ω: ∇v(x)=ζ_j})=t_jL^ν(ω)for everyj∈ {0, . . . , ν}.

Proof. Let us setC=co({ζ₀, . . . , ζ_ν}). ThenC is a bounded polyhedral set. Moreover, by (1.2), 0∈int(C), and, by Proposition 1.1,C^◦turns out to be polyhedral, compact, and 0∈int(C^◦).

By Proposition 1.8 applied ton=ν,Ω=ω, andK=C^◦, there exist{x_k} ⊆ωand{t_k} ⊆ ]0,1]such that the sets {x_k+t_kC^◦:k∈N}are contained inω, pairwise disjoint, andL^ν(ω\

k∈N(x_k+t_kC^◦))=0.

Letσ_Cbe the support function ofC. Then it is easy to verify that σ_C(x)=max

u_ζj(x): j∈ {0, . . . , ν}

for everyx∈R^ν, from which it follows thatσ_C∈PA(R^ν).

For everyk∈Nandx∈x_k+t_kC^◦, we now define w_k(x)=t_kσ_C

x−x_k t_k

−t_k.