A characterization of convex calibrable sets in <span class="mathjax-formula">${R}^{N}$</span> with respect to anisotropic norms

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A characterization of convex calibrable sets in R ^N with respect to anisotropic norms

V. Caselles

, A. Chambolle

, S. Moll

, M. Novaga

aDepartament de Tecnologia, Universitat Pompeu-Fabra, Barcelona, Spain bCMAP, Ecole Polytechnique, CNRS, Palaiseau, France cDipartimento di Matematica, Università di Pisa, Pisa, Italy

Received 1 January 2005; accepted 22 April 2008 Available online 7 May 2008

Abstract

A set is called “calibrable” if its characteristic function is an eigenvector of the subgradient of the total variation. The main purpose of this paper is to characterize the “φ-calibrability” of bounded convex sets in R^N with respect to a normφ (called anisotropyin the sequel) by the anisotropic meanφ-curvature of its boundary. It extends to the anisotropic and crystalline cases the known analogous results in the Euclidean case. As a by-product of our analysis we prove that any convex bodyCsatisfying aφ-ball condition contains a convexφ-calibrable setKsuch that, for anyV ∈ [|K|,|C|], the subset ofCof volumeV which minimizes the φ-perimeter is unique and convex. We also describe the anisotropic total variation flow with initial data the characteristic function of a bounded convex set.

©2008 Elsevier Masson SAS. All rights reserved.

Résumé

On dit qu’un ensemble est « calibrable » si sa fonction est vecteur propre du sous-gradient de la variation totale. Le but de cet article est une caractérisation de la «φ-calibrabilité » des ensembles convexes bornés deR^N, relativement à une normeφ(appelée anisotropie), en fonction de laφ-courbure moyenne anisotrope de leur frontière. Il s’agit donc d’une extension aux cas anisotropes et cristallins de résultats connus dans le cas euclidien. On démontre en particulier l’existence dans tout corps convexe régulierC d’un convexeK⊆C φ-calibrable, tel que pour toutV∈ [|K|,|C|], l’ensemble de volumeV deφ-périmètre minimal contenu dans Cest unique et convexe. Nous étudions aussi le flot de la variation totale anisotrope à partir de la caractéristique d’un ensemble convexe borné.

©2008 Elsevier Masson SAS. All rights reserved.

MSC:35J70; 49J40; 52A20; 35K65

Keywords:Calibrable sets; Convex sets; Mean curvature; Total variation

Mots-clés :Ensembles calibrables ; Ensembles convexes ; Courbure moyenne ; Variation totale

* Corresponding author.

E-mail addresses:vicent.caselles@upf.edu (V. Caselles), antonin.chambolle@polytechnique.fr (A. Chambolle), salvador.moll@upf.edu (S. Moll), novaga@dm.unipi.it (M. Novaga).

0294-1449/$ – see front matter ©2008 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2008.04.003

1. Introduction

The purpose of this paper is to give a characterization of convex calibrable sets (with respect to an anisotropic perimeter) in R^N extending the corresponding result for N =2 [17] and the corresponding results for the usual euclidean perimeter [27,14,2]. In the evolution of a set under anisotropic mean curvature flow, calibrable facets are those which do not bend or break during the evolution process, and they are characterized, in the convex case, in terms of the anisotropic curvature of the boundary [17].

The anisotropic perimeterP_φinR^Nis defined as P_φ(E):=

∂E

φ^◦ ν^E

dH^N−1, E⊆R^N,

whereν^Eis the outward unit normal to the boundary∂EofEandφ^◦(the surface tension) is a norm onR^N. We say that the anisotropyφ^◦iscrystallineif{φ^◦1}is a polyhedron.

LetF be a convex subset ofR². For any measurable setX⊆R^N,|X|denotes the Lebesgue measure of the setX.

It has been proved in [17] that the following three assertions are equivalent.

(a) F isφ-calibrable, i.e., there is a vector field ξ ∈L^∞(F,R²), withφ(ξ(x))1 a.e. inF (where φis the dual norm ofφ^◦), such that

−divξ=λ^φ_F :=P_φ(F )

|F| inF, ξ·ν^F= −φ^◦

ν^F

in∂F, (1.1)

whereν^F(x)denotes the outer unit normal to∂F at the pointx∈∂F. (b) F is a solution of the problem

Xmin⊆FP_φ(X)−λ^φ_F|X|. (1.2)

(c) We have ess sup

x∈∂F

κ_F^φ(x)λ^φ_F, (1.3)

whereκ_F^φ(x)denotes the anisotropic curvature of∂F at the pointx.

The characterization of the calibrability of a convex set inR², with respect to the euclidean perimeter, was proved by Giusti in [27], where he also proved that in a convex calibrable set the capillary problem in absence of gravity, with any prescribed contact angle at its boundary, has always a solution. In the euclidean case, this equivalence has been partly rederived in [14] where calibrable sets were used to construct explicit solutions of the denoising problem in image processing. A simple proof of the equivalence(b)⇔(c)was given in [30] (where it was studied in connection which Cheeger sets, see Section 6). The extension of the above result for the euclidean perimeter and N3 was proved in [2]. In that case, the left-hand side of (1.3) has to be substituted by the sum of the principal curvatures at the pointx∈∂F. Our purpose in this paper is to extend the above set of equivalences to the anisotropic case, for a convex set inR^Nwhich satisfies a ball condition (see Definition 2.7).

The proof of the equivalence(a)⇔(b)is the same as in the euclidean case and it is independent of the dimensionN (see [14,2]). We notice that the supremum of the curvatureκ_C^φ in (1.3) has to be substituted with the number(N− 1)H^φ_C∞, whereH^φ_C∞is defined in Section 2.5 and denotes theL^∞-norm of the anisotropic mean curvature of

∂C. To prove(b)⇔(c)we follow the strategy used in [2] for the euclidean case, thus, we embed the variational problem (1.2) in a family of problems

Xmin⊆CP_φ(X)−λ|X|, λ >0, (1.4)

and we study the dependence of its solution onλ. In particular, we prove thatCis a solution of (1.4) if and only if λmax{λ^φ_C, (N−1)H^φ_C∞}. The solutions of (1.4) are related to the solution of the variational problem

min

u∈BV(R^N)∩L²(R^N)

R^N

φ^◦(Du)+μ 2

R^N

(u−χ_C)²dx, μ >0. (1.5)

Indeed, it turns out that the level sets of the solution of (1.5) embed the solutions of (1.4) forλ∈ [0, μ]. Since the solutionuof (1.5) satisfies the equation

v−μ⁻¹div

∂φ^◦(Dv)

=1 inC,

∂φ^◦(Dv)·ν^C= −φ^◦ ν^C

in∂C (1.6)

(the meaning of∂φ^◦(Dv)will be explained below) and the solutions of (1.6) can be approximated by the solutionsu of

v−μ⁻¹div

T^◦(Dv) 2+φ^◦(Dv)²

=1 inC, T^◦(Dv)

2+φ^◦(Dv)²·ν^C= −φ^◦ ν^C

in∂C (1.7)

as →0 (whereT^◦(x)= ¹₂∂(φ^◦)²(x),x∈R^N). We use the result of Korevaar [31] to conclude thatuis concave inC, hence also continuous there. This implies the uniqueness and convexity of solutions of (1.4). Thus, by studying the dependence onλ of solutions of (1.4), we can prove that ifC satisfies the curvature estimate (1.3) but is not a minimum of (1.2), then it can be approximated from inside by solutionsC_λof (1.4), withλ→μandμ > λ^φ_C. As we shall prove in Proposition 7.1, this implies that(N−1)H^φ_C∞> λ^φ_C, a contradiction.

As an interesting by-product of our analysis we obtain that solutions of (1.4) are convex sets. Since (1.4) can be considered as the functional obtained by applying the Lagrange multiplier method to the area minimizing problem

X⊆C,min|X|=VP_φ(X) (1.8)

where 0< V <|C|, we obtain that, for some range of volumes, the solutions of this isoperimetric problem with fixed volumeV are convex sets. The range of values of V for which the above result holds is[|K|,|C|] whereK is a convexφ-calibrable set contained inC obtained as solution of (1.4) for a certain value ofλ(see Section 6). This extends the analogous result in [2]. In the euclidean case, a similar result has been also proved by E. Stredulinsky and W.P. Ziemer [39] in the case of a convex setC containing a ballB such that∂B∩∂Cis a meridian of∂B, and we mention the result of C. Rosales [36] whenC is a rotationally symmetric convex body.

Finally, let us mention that our results enable us to describe the evolution of any convex set inR^N, satisfying a ball condition, by the anisotropic total variation flow. The same result for the euclidean case was proved in [3] (forN=2) and in [2]: as in those papers, it can be extended to unions of convex set which are far apart from each other. Other examples of evolution are given in [35].

Let us describe the plan of the paper. In Section 2 we collect some preliminary definitions and results about anisotropies, regularity conditions in the anisotropic case, functions of bounded variation and Green’s formula. In Section 3 we recall the subdifferential of the anisotropic total variation inR^N and we defineφ-calibrable sets. In Section 4 we relate the solution of the variational problem (1.4) with the solution of (1.5) and we study the basic properties of its minimizers. In Section 5 we prove the concavity of solutions of (1.5) for a certain range of values of μ. This will imply the convexity of the solutions of (1.4) for an interval of values of λ. In Section 6 we prove the convexity of solutions of (1.8) whenV ∈ [|K|,|C|]whereK is a certain convexφ-calibrable set contained inC. Section 7 is devoted to the characterization of theφ-calibrability of a convex set in terms of the anisotropic mean curvature of its boundary. Finally, in Section 8 we characterize theφ-calibrability of the convex sets which satisfy a ball condition, and we describe the evolution of such sets by the minimizing anisotropic total variation flow.

2. Preliminaries 2.1. Notation

Given an open set A⊆R^N and a functionf:A→R, we writef ∈C^1,1(A)(resp.f ∈C_loc^1,1(A)) if f ∈C¹(A) and∇f ∈Lip(A;R^N)(resp.∇f ∈Lip_loc(A;R^N)). LetB⊂R^N be a set; we say that B (or ∂B) is of class C^1,1 (resp. Lipschitz) if∂B can be written, locally around each point, as the graph (with respect to a suitable orthogonal coordinate system) of a functionf of classC^1,1(resp. Lipschitz).

Given two nonempty sets A, B, we denote the Hausdorff distance between A and B by d_H(A, B) = max{supa∈Adist(a, B),sup_b∈Bdist(b, A)}. We denote byχ_Athe characteristic function ofA, and byA¯(resp. int(A)) the closure (resp. the interior part) ofA.

We let S^N−1:= {ξ ∈R^N: |ξ| =1}and for ρ >0 we let Bρ := {x ∈R^N: |x|< ρ}. We denote by H^N−1 the (N−1)-dimensional Hausdorff measure inR^N, and by| · |the Lebesgue measure. Given a functionf defined on the boundary∂Cof a setC, we setfL^∞(∂C)to be theH^N−1-essential supremum of|f|on∂C.

We shall use the notationf (t )∈O(t )if|^{f (t )}_t |is bounded ast→0.

2.2. Anisotropies and distance functions

In the sequel of the paper, the functionφwill always denote an anisotropy, i.e., a functionφ:R^N→ [0,∞)such that

φ(tξ )= |t|φ(ξ ) ∀ξ∈R^N, ∀t∈R, (2.1)

and

m|ξ|φ(ξ ) ∀ξ∈R^N, (2.2)

for some m >0. In particular φ(ξ )=φ(−ξ ) for any ξ ∈R^N. Observe that there existsM∈ [m,+∞)such that φ(ξ )M|ξ| for allξ ∈R^N. We let Wφ:= {φ1}. The polar function φ^◦ of φ (also called surface tension) is defined asφ^◦(ξ ):=sup{η·ξ:φ(η)1}for anyξ∈R^N. Ifφis an anisotropy, thenφ^◦is also an anisotropy and there holds(φ^◦)^◦=φ.

By a convex body we mean a compact convex set whose interior contains the origin. A convex body is said to be centrally symmetric if it is symmetric with respect to the origin. Ifφis an anisotropy, thenWφ:= {ξ: φ(ξ )1} (sometimes called Wulff shape) is a centrally symmetric convex body. IfKis a convex body, the functionh_K(ξ ):=

sup_η∈Kη·ξ is called the support function ofK; notice that{(h_K)^◦1} =K.

As usual, we shall denote by∂φ(ξ )the subdifferential ofφatξ∈R^N. Ifφis differentiable atξ, we have∂φ(ξ )= {∇φ(ξ )}. IfΦis a convex function defined on a Hilbert space, we still denote by∂Φthe subdifferential ofΦ.

Given a nonempty setE⊆R^N, we let d_φ(x, E):= inf

y∈Eφ(x−y), x∈R^N.

We denote byd_φ^Ethe signedφ-distance function to∂Enegative insideE, that is d_φ^E(x):=d_φ(x, E)−d_φ

x,R^N\E

, x∈R^N. (2.3)

Observe that|d_φ^E(x)| =dφ(x, ∂E).

It can be shown (the proofs are identical to the Euclidean case) that the functiond_φ^Eis Lipschitz, and at each point x where it is differentiable we haveφ^◦(∇d_φ^E(x))=1. We set

ν_φ^E:= ∇d_φ^E on∂E, (2.4)

at those points where∇d_φ^Eexists. Whenφis the euclidean norm, i.e.,φ(ξ )= |ξ|, we setν^E=ν_|·|^EandB1=W|·|. We have

ν_φ^E(x)= ν^E(x) φ^◦(ν^E(x)).

LetT^◦be the multivalued map inR^Ndefined by T^◦(x)=1

2∂ φ^◦2

(x), x∈R^N.

T^◦is a maximal monotone operator mappingWφ^◦ontoWφ. IfEis Lipschitz, atH^N−1-a.e.x∈∂Ewe have ν_φ^E(x), p =1 ∀p∈T^◦

ν_φ^E(x) .

Vector fields which are selections in∂φ^◦(∇d_φ^E)are sometimes called Cahn–Hoffman vector fields, and we denote by Norφ(∂E,R^N)the set of such fields.

Definition 2.1.We say thatφ∈C₊^1,1(resp.C₊^∞) ifφ²is of classC^1,1(R^N)(resp.C^∞(R^N\ {0})) and there exists a constantc >0 such that∇²(φ²)cId almost everywhere. We say that a centrally symmetric convex body is of class C₊^1,1(resp.C₊^∞) if it is the unit ball of an anisotropy of classC₊^1,1(resp.C₊^∞).

Definition 2.2.We say thatφis crystalline if the unit ballWφofφis a polytope.

Remark 2.3.Observe that

(a) φ∈C₊^1,1(resp.C₊^∞) if and only ifφ^◦∈C₊^1,1(resp.C₊^∞) [37, p. 111];

(b) φis crystalline if and only ifφ^◦is crystalline.

2.3. φ-regularity and theRWφ-condition

Following [15–18] we define the class ofφ-regular sets and Lipschitzφ-regular sets (these latter are a generaliza- tion of sets of classC^1,1in the euclidean case).

Definition 2.4.LetE⊂R^Nbe a set. We say thatEisφ-regular if∂Eis a compact Lipschitz hypersurface and there exist an open setU⊃∂Eand a vector fieldn∈L^∞(U;R^N)such that divn∈L^∞(U ), andn∈∂φ^◦(∇d_φ^E)almost everywhere inU. We say thatEis Lipschitzφ-regular ifEisφ-regular andn∈Lip(U;R^N).

It is clear that a Lipschitzφ-regular set isφ-regular. With a little abuse of notation, sometimes we will denote by (E, n), by(E, U )or by(E, U, n), aφ-regular set.

Observe that, in general, vector fieldsnare not unique, unlessφ∈C₊^1,1. Whenφ∈C₊^1,1the inclusionn∈∂φ^◦(∇d_φ^E) becomes an equality; in this respect we give the following definition.

Definition 2.5.Letφ∈C₊^1,1and(E, U )be a Lipschitzφ-regular set. Letx∈Ube a point where there exists∇d_φ^E(x).

We set

n^E_φ(x):= ∇φ^◦

∇d_φ^E(x)

. (2.5)

Remark 2.6.Observe that(Wφ, n), withn(x):=x/φ(x), is Lipschitzφ-regular, and divn(x)=(N−1)/φ(x)for almost everyx∈R^N.

The next definition will play an important role in the sequel.

Definition 2.7.LetE⊂R^N be a set with nonempty interior andR >0. We say thatEsatisfies theRWφ-condition if, for anyx∈∂E, there existsy∈R^N such that

RWφ+y⊆ ¯E and x∈∂(RWφ+y).

The first assertion of the following result is proved in [18, Lemmas 3.4, 3.5], and the second one is proved in [13, Proposition 3.9].

Lemma 2.8.Letφbe any anisotropy.

(i) IfEis a Lipschitzφ-regular set, thenEandR^N\Esatisfy theRWφ-condition for someR >0.

(ii) A compact convex set satisfying theRWφ-condition isφ-regular.

Ifφ∈C₊^1,1, we list some relations betweenφ-regularity and theRWφ-condition (see [13, Remark 4]).

Remark 2.9.Assume thatφ∈C₊^1,1. The following assertions hold.

(a) Eis Lipschitzφ-regular if and only ifEis of classC^1,1.

(b) LetCbe a compact convex set which satisfies theRWφ-condition for someR >0. ThenCis Lipschitzφ-regular (henceCis of classC^1,1by (a)).

(c) Eis Lipschitzφ-regular if and only ifEandR^N\Esatisfy theRWφ-condition for someR >0.

2.4. BV functions,φ-total variation and generalized Green formula

LetΩbe an open subset ofR^N. A functionu∈L¹(Ω)whose gradientDuin the sense of distributions is a (vector valued) Radon measure with finite total variation|Du|(Ω)inΩ is called a function of bounded variation. The class of such functions will be denoted byBV(Ω). We denote byBVloc(Ω)the space of functionsw∈L¹_loc(Ω)such that wϕ∈BV(Ω)for allϕ∈Cc^∞(Ω). Concerning all properties and notation relative to functions of bounded variation we will follow [6].

A measurable set E⊆R^N is said to be of finite perimeter in Ω if |Dχ_E|(Ω) <∞. The (euclidean) perimeter ofEinΩ is defined asP (E, Ω):= |Dχ_E|(Ω), and we haveP (E, Ω)=P (R^N\E, Ω). We shall use the notation P (E):=P (E,R^N).

Letu∈BV(Ω). We define the anisotropic total variation ofuwith respect toφinΩ [4] as

Ω

φ^◦(Du)=sup

Ω

udivσ dx: σ∈Cc¹

Ω;R^N , φ

σ (x)

1∀x∈Ω

. (2.6)

IfE⊆R^Nhas finite perimeter inΩ, we set P_φ(E, Ω):=

Ω

φ^◦(Dχ_E)

and we have [4]

P_φ(E, Ω)=

Ω∩∂^∗E

φ^◦ ν^E

dH^N−1, (2.7)

where∂^∗Eis the reduced boundary ofEandν^Ethe (generalized) outer unit normal toEat points of∂^∗E.

Recall that, sinceφ^◦is homogeneous,φ^◦(Du)coincides with the nonnegative Radon measure inR^N given by φ^◦(Du)=φ^◦

∇u(x)

dx+φ^◦ D^su

|D^su|

D^su,

where∇u(x) dxis the absolutely continuous part ofDu, andD^suits singular part.

LetΩ be an open subset ofR^N. Following [10], let X₂(Ω):=

z∈L^∞ Ω;R^N

: divz∈L²(Ω) .

Ifz∈X2(Ω)andw∈L²(Ω)∩BV(Ω)we define the distribution(z, Dw):Cc^∞(Ω)→Rby the formula (z, Dw), ϕ := −

Ω

wϕdivz dx−

Ω

wz· ∇ϕ dx ∀ϕ∈C_c^∞(Ω).

Then(z, Dw)is a Radon measure inΩ,

Ω

(z, Dw)=

Ω

z· ∇w dx ∀w∈L²(Ω)∩W^1,1(Ω), and

B

(z, Dw)

B

(z, Dw)z∞

B

|Dw| ∀B⊆Ω Borel set.

We recall the following result proved in [10].

Theorem 2.10.LetΩ⊂R^Nbe a bounded open set with Lipschitz boundary. Letu∈BV(Ω)∩L²(Ω)andz∈X2(Ω).

Then there exists a function[z·ν^Ω] ∈L^∞(∂Ω)such that[z·ν^Ω]L^∞(∂Ω)zL^∞(Ω;R^N), and

Ω

udivz dx+

Ω

(z, Du)=

∂Ω

z·ν^Ω

u dH^N−1.

WhenΩ =R^N we have the following integration by parts formula [10], forz∈X2(R^N) andw∈L²(R^N)∩ BV(R^N):

R^N

wdivz dx+

R^N

(z, Dw)=0. (2.8)

Remark 2.11.LetΩ⊂R^Nbe a bounded Lipschitz open set, and letz_inn∈L^∞(Ω;R^N)with divz_inn∈L²_loc(Ω), and z_out∈L^∞(R^N\Ω;R^N)with divz_out∈L²_loc(B_R\Ω), for allR >0. Assume that

zinn·ν^Ω

(x)= −

zout·ν^R2\Ω

(x) forH^N−1−a.ex∈∂Ω.

Then if we definez:=z_innonΩ andz:=z_outonR^N\Ω, we havez∈L^∞(R^N;R^N)and divz∈L²_loc(R^N).

2.5. The anisotropic mean curvature

Let(E, U, n)be aφ-regular set. For anyp∈ [1,+∞], we define H˜_φ^div,p

U,R^N :=

N∈L^∞ U;R^N

: N∈T^◦

∇d_φ^E

,divN∈L^p(U ) .

Fix nowδ₀>0 be such thatU_t:= {|d_φ^E|< t} ⊆Ufort∈ [0, δ0]. Then, following [18] (see also Theorem 2.12 below) there exists a vector fieldz˜t∈L^∞(Ut,R^N)such thatz˜t∈T^◦(∇d_φ^E)a.e. inU0, divz˜t ∈L²(U0)and

divz˜_tL²(Ut)divZL²(Ut) ∀Z∈ ˜H_φ^div,2 U_t,R^N

. (2.9)

We point out that, even if the minimizerz˜_tmay be nonunique, its divergence is always uniquely defined. In particular, it follows that

divz˜_s=divz˜_t a.e. inU_s, (2.10)

for all 0< s < t.

Theorem 2.12.Let(E, U, n)be aφ-regular set. Let0< δ₀Rbe such thatU₀:= {|d_φ^E|< δ₀} ⊆U, and let(u^h, z^h), u^h∈BVloc(R^N)∩L²_loc(R^N), be the solution of

u^h−hdivz^h=d_φ^E inR^N, (2.11)

wherez^h∈∂φ^◦(∇u^h)and(z^h, Du^h)=φ(Du^h)inD(R^N). Then, there existsz˜∈L^∞(R^N,R^N), and a subsequence h_j→0⁺such thatz^hj→ ˜zweakly^∗inR^N, wherez˜is such thatz˜∈T^◦(∇d_φ^E)inU₀and

divz˜L^q(U0)divnL^q(U0) ∀q∈ [1,∞]. (2.12)

More generally,z˜satisfies the following inequality divz˜L^q(U_δ)divZL^q(U_δ) ∀Z∈ ˜H_φ^div,∞

Uδ,R^N

, (2.13)

for allq∈ [1,∞]and for all0< δ < δ0, whereUδ:= {|d_φ^E|< δ}. Finally, ifEis convex, thendivz˜0inU0. Let us recall that (2.11) has a unique solutionu^h∈L²_loc(R^N)[23]. Moreoveru^h∈L^∞_loc(R^N)[23] andu^hL^∞(B_R) d_φ^EL^∞(B2R)+Cfor some constantCwhich does not depend onh. Let us also point out thatu^his Lipschitz with a Lipschitz constant depending only on the Lipschitz constant ofd_φ^E. Indeed, by the results in [23]u^hcan be obtained as limit inL¹_loc(R^N)of the solutionsu^h_j∈L^∞_loc(R^N)of

u−hdiv∂φ(∇u)min d_φ^E, j

inR^N, (2.14)

and, for any y∈R^N,u^h_j(· +y)is the solution of (2.14) with right-hand side min{d_φ^E(· +y), j}. As in [23], Corol- lary C.2, we prove that

u^h_j−u^h_j(· +y)₊

∞min d_φ^E, j

−min

d_φ^E(· +y), j_∞d_h^E−d_h^E(· +y)

∞. This implies that

u^h−u^h(· +y)₊

∞d_h^E−d_h^E(· +y)

∞. Interchanging the role ofu^handu^h(· +y)we deduce that

∇u^h

∞∇d_φ^E

∞. (2.15)

We may also prove this along the lines of the proof of Theorem 3 in [23] which uses another approximation of (2.11) and viscosity solution theory.

Proof. For simplicity, let us denoted:=d_φ^E. By the remarks previous to the proof we have that|u^h|cλon{dλ} wherec_λis a constant depending onλfor anyλ >0. Multiplying (2.11) byu^h−dand integrating by parts in{dλ} we obtain

{dλ}

u^h−d2

dx= −h

{dλ}

z^h·

∇u^h− ∇d dx+h

∂{dλ}

z^h·ν^{dλ} u^h−d

dH^N−1,

henceu^h→dinL²_loc(R^N)ash→0⁺. By the estimate (2.15), we have that the convergence takes place also locally uniformly inR^N. Moreover, modulo a subsequence, we may assume thatz^h→ ˜zweakly^∗ inL^∞(R^N)ash→0⁺. Let a < bandQ^h_a,b:= {u^ha} ∩ {db}be such thatQ^h_a,b⊆U₀. Let us assume thath varies along a sequence converging to 0. Sinceu^h∈BVloc(R^N)we may assume thatais such that{u^h< a}is a set of finite perimeter inR^N. Since u^h converges tod locally uniformly inR^N we may assumehsmall enough so that{u^h< a} ⊆ {d b}and {u^h=a} ∩ {d=b} = ∅. LetP:R→ [0,∞)be a smooth, increasing and nonnegative function. Then

Q^h_a,b

u^h−d P

u^h−d dx=h

Q^h_a,b

divz^hP u^h−d

dx

=h

Q^h_a,b

divz^h−divn P

u^h−d dx+h

Q^h_a,b

divnP u^h−d

dx.

The first term can be written as

Q^h_a,b

divz^h−divn

P (u^h−d) dx

= −

Q^h_a,b

z^h−n

· ∇P u^h−d

dx−

R^N

z^h, Dχ_Qh a,b

−(n, Dχ_Qh a,b)

P u^h−d

.

First, observe that

Q^h_a,b

z^h−n

· ∇P u^h−d

dx=

Q^h_a,b

P

u^h−d z^h−n

· ∇ u^h−d

dx

=

Q^h_a,b

P

u^h−d φ^◦

∇u^h

−n· ∇u^h+φ^◦(∇d)−z^h· ∇d dx0.

To prove that the second term is negative, we observe that

−

R^N

z^h, Dχ_Qh a,b

−(n, Dχ_Qh a,b)

P u^h−d

=

R^N

z^h, Dχ_{uh<a}

−(n, Dχ_{uh<a}) P

u^h−d

−

R^N

z^h, Dχ_{db}

−(n, Dχ_{db}) P

u^h−d .

Now, by the proof of [23, Lemma 5.1] (see also [13, Lemma 4]), we have that−(z^h, Dχ_{uh<s})=φ^◦(Dχ_{uh<s}), where the equality means the equality of both measures, for almost everys∈Rand we may assume thatahas been chosen to satisfy this equality. On the other hand, sinceφ(n)1, we have that|(n, Dχ_{uh<a})|φ^◦(Dχ_{uh<a}). This implies that

R^N

z^h, Dχ_{uh<a}

−(n, Dχ_{uh<a}) P

u^h−d

0.

By the same arguments we could have also chosenb > afrom the beginning so that(n, Dχ_{db})= −φ^◦(Dχ_{db}), and, again, we have|(z^h, Dχ_{db})|φ^◦(Dχ_{db}). Hence

R^N

z^h, Dχ_{db}

−(n, Dχ_{db}) P

u^h−d

0.

Combining all these inequalities we obtain that

Q^h_a,b

u^h−d P

u^h−d dxh

Q^h_a,b

divnP u^h−d

dx. (2.16)

If q <∞, let q˜ =q. If q = ∞, let q <˜ ∞. Let Pj be a sequence of increasing nonnegative functions such that P_j(r)→r^+(q˜−1)locally uniformly asj → ∞. UsingP=P_jin (2.16) we obtain

1 h

Q^h_a,b

u^h−d₊q_˜

dx

Q^h_a,b

divn

u^h−d₊q_˜−1

dx.

Applying Young’s inequality we obtain 1

hu^h−d₊

L^q˜(Q^h_a,b)divn_Lq˜(Q^h_a,b). Hence, we have

divz^h₊

_Lq˜(Q^h_a,b)divn_Lq˜(Q^h_a,b). Lettingh→0 andq˜→ ∞ifq= ∞, we obtain

(divz)˜ ⁺

L^q(Qa,b)divnL^q(Q_a,b) ∀q∈ [1,∞], (2.17)

whereQ_a,b:= {adb}. Lettinga→ −δ₀,b→δ₀, we deduce that (divz)˜ ⁺

L^q(U₀)divnL^q(U0) ∀q∈ [1,∞]. (2.18)

I a similar way we obtain (divz)˜ ⁻

L^q(U₀)divnL^q(U0) ∀q∈ [1,∞]. (2.19) Indeed it suffices to changeu^h into−u^h,n into−nand to integrate in {u^hb} ∩ {da}to obtain (2.19). Both inequalities (2.18) and (2.19) prove (2.12).

Now, we observe thatu^h→d locally uniformly inR^N,z^h→ ˜zand divz^h→divz˜weakly inL²_loc(U0). From this it follows thatz(x)˜ ∈∂T^◦(∇d)a.e. inU₀. To prove it, observe that sinceφ(z^h)1 we deduce thatφ(z)˜ 1. Letψ be a nonnegative test function with support contained inU₀. Then

U0

φ^◦(∇d)ψ dxlim inf

h→0

U0

φ^◦

∇u^h

ψ dx=lim inf

h→0

U0

z^h· ∇u^hψ dx

=lim inf

h→0 −

U₀

divz^hu^hψ dx−

U₀

z^h· ∇ψ u^hdx

= −

U₀

divz dψ dx˜ −

U₀

˜

z· ∇ψ d dx

=

U₀

˜

z· ∇dψ dx

U₀

φ^◦(∇d)ψ dx.

Hence

U0

˜

z· ∇dψ dx=

U0

φ^◦(∇d)ψ dx.

Since this is true for any test functionψ with compact support inU0we obtain thatz˜· ∇d=φ^◦(∇d)inU0, hence

˜

z∈T^◦(∇d)inU0.

To prove the inequality (2.13) we observe that if 0< δ < δ0andZ∈ ˜H_φ^div,∞(Uδ,R^N), then(E, Uδ, Z)isφ-regular and, by repeating the computations that lead to (2.12), we deduce that (2.13) holds.

Finally, ifEis convex, the inequality divz˜0 follows from the inequalitydu^h, proved in [23, Theorem 3]. 2 From (2.10) and (2.13) it follows that, if E satisfies the assumptions of Theorem 2.12, the function t → divz˜_tL^∞(Ut)= divz˜L^∞(Ut)is nondecreasing, hence we may take the limit

H^φ_E

∞:= lim

t→0⁺divz˜tL^∞(U_t). (2.20)

Let(E, n)be Lipschitzφ-regular and let N∈Norφ(∂E,R^N)∩lip(∂E,R^N). By [18, Lemmas 3.4, 3.5, 4.5], we have that

(i) there exists a neighborhoodUof∂Eandδ >0 such that the mapF_N:∂E×(−δ, δ)→R^N defined by F_N(x, t )=x+tN(x)

is bilipschitz, moreover d_φ^E

x+tN(x)

=t, x∈∂E,

and∇d_φ^E(x+tN(x))=ν_φ^E(x)for anyt∈(−δ, δ)andH^N−1-a.e.x∈∂E;

(ii) giveny∈U, there is a uniquex ∈∂Esuch thaty=F_N(x, t )wheret=d_φ^E(x). We shall denote this pointx byπ_N(y). This permits to extend the vector field N to a vector field N^eonU by the formula

N^e(x)=N π_N(x)

, x∈U.

Usingπ_N, any vector fieldηcan be extended from∂EtoU. Hence, from now on we shall writeηinstead ofη^e, i.e. we shall assume thatηis defined on a neighborhood of∂E;

(iii) the trace of div N^e(denoted by div N) is definedH^N−1-almost everywhere on∂Eand coincides on∂Ewith the tangential divergence of N to be defined below.

Finally, if(E, n)is a Lipschitzφ-regular set and N∈Norφ(∂E,R^N), we may define the (weak) tangential diver- gence divτN : Lip(∂E)→Ras follows

∂E

divτNψ φ^◦ ν^E

dH^N−1:=

∂E

N·nψdivτnφ^◦ ν^E

dH^N−1−

∂E

(Id−n⊗n)∇τψ

·Nφ^◦ ν^E

dH^N−1,

whereψ∈Lip(∂E). As proved in [18], this divergence does not depend on the vector fieldn. Letting H_φ^div,p

∂E,R^N :=

N∈Norφ

∂E,R^N

: divτN∈L^p(∂E)

, p∈ [1,+∞], we define Nmin∈H_φ^div,2(∂E,R^N)to be a minimizer (possibly nonunique) of the functional

∂E

(divτN)²φ^◦ ν^E

dH^N−1, N∈H_φ^div,2

∂E,R^N

. (2.21)

As proved in [18], the function divτ Nmindoes not depend on the choice of the minimizer Nminof (2.21). Moreover, by [18, Theorem 6.7] we have that divτNmin∈L^∞(∂E)and

divτNmin∞=min

divτN∞: N∈H_φ^div,∞

∂E,R^N

. (2.22)

Remark 2.13.Letφ∈C₊^1,1andEbe a Lipschitzφ-regular set. Then

divτNmin=divn^E_φ, H^N−1-a.e. on∂E and (N−1)H^φ_E∞= divτNminL^∞(∂E). (2.23) We do not know if the second equality in (2.23) holds for all Lipschitzφ-regular setE⊂R^N. However, we can prove it under the additional assumption that the anisotropyφis crystalline andEis a polyhedron.

Let us first observe that a polyhedronE⊂R^N is Lipschitzφ-regular if and only if for all verticesv ofE there holds

C(v):=

F facet ofE:v∈F

∂φ^◦ ν^F

= ∅, (2.24)

whereν^F is the outer unit normal to∂Eat the facetF.

Proposition 2.14.Assume thatφis crystalline and letE⊂R^Nbe a Lipschitzφ-regular polyhedron. Then (N−1)H^φ_E

∞= divτNminL^∞(∂E).

Proof. Given a vertexvofE, we shall denote byN (v)a generic element of the setC(v), defined by (2.24).

LettingE_t:= {d_φ^Et}, we know from [18] that there existsδ₀>0 such thatE_tis a Lipschitzφ-regular polyhedron for all|t|δ₀. Let also N^t_min:∂E_t→R^N be a minimizer of divτNL²(∂Et), which is equivalently a minimizer of divτNL^∞(∂E_t)by [18]. LettingH_t:= divτN^t_minL^∞(∂E_t), it is enough to prove that the functiont∈ [−δ₀, δ₀] →H_t is continuous att=0 (hence it is also continuous on the whole interval). Indeed, lettingz˜ as in Theorem 2.12 and differentiating the equalityφ(z)˜ =1, we obtain∇ ˜z· ∇d_φ^E=0 in a neighborhood of∂E. As a consequence, we get that divτz˜=divz˜ a.e. in that neighborhood, where the tangential divergence (which, in this case, is an euclidean divergence) is computed with respect to∂E_tat a pointx∈∂E_t. It follows that the fieldz˜can be obtained by patching together the minimizing vector fields N^t_min, which are defined on∂E_t.

Letting nowF_t be the facet ofE_t corresponding to the facetF ofE, we shall prove the equivalent statement that the function

t→H_t^F:=divτN^t_min

L^∞(Ft)