Γ-Limits of convolution functionals

Γ-limits of convolution functionals

Luca Lussardi and Annibale Magni

Preprint 2010-11 November 2010

Fakult¨ at f¨ ur Mathematik

Technische Universit¨ at Dortmund Vogelpothsweg 87

44227 Dortmund tu-dortmund.de/MathPreprints

Γ-LIMITS OF CONVOLUTION FUNCTIONALS

LUCA LUSSARDI AND ANNIBALE MAGNI

Abstract. We compute the Γ-limit of a sequence of non-local integral functionals de- pending on a regularization of the gradient term by means of a convolution kernel. In particular, as Γ-limit, we obtain free discontinuity functionals with linear growth and with anisotropic surface energy density.

Keywords: Free discontinuities, Γ-convergence, anisotropy.

2010 Mathematics Subject Classification: 49Q20, 49J45, 49M30.

Contents

1. Introduction 1

2. Notation and preliminaries 4

2.1. Functions of bounded variation 4

2.2. Slicing 5

2.3. Γ-convergence 5

2.4. Supremum of measures 6

2.5. A density result 6

2.6. A relaxation result 6

3. Statement of the main results 7

3.1. The anisotropy 8

3.2. Main results 9

4. Compactness 10

5. The Γ-liminf inequality 11

5.1. A preliminary estimate from below in terms of the volume and Cantor parts 11

5.2. A preliminary estimate in terms of the surface part 15

5.3. Proof of the Γ-liminf inequality 19

6. The Γ-limsup inequality 21

7. Computation of θin the one-dimensional case 24

References 26

1. Introduction

As it is well known, many variational problems which are recently under consideration, arising for instance from image segmentation, signal reconstruction, fracture mechanics and liquid crystals, involve afree discontinuity set (according to a terminology introduced in [19]). This means that the variable functionuis required to be smooth outside a surfaceK, depending onu, and bothu andK enter the structure of the functional, which takes the form given by

F(u, K) = Z

Ω\K

φ(|∇u|) dx+ Z

K∩Ω

θ(|u⁺−u⁻|, νK) dHⁿ⁻¹,

being Ω an open subset ofRⁿ, K is a (n−1)-dimensional compact subset ofRⁿ, |u⁺−u⁻| the jump ofuacrossK,νKthe normal direction toK, whileφandθgiven positive functions, whereas Hⁿ⁻¹ denotes then−1-dimensional Hausdorff measure.

1

The classical weak formulation for such problems can be obtained consideringK as the set of the discontinuities ofuand thus working in the space of functions with bounded variation. More precisely, the aforementioned weak form ofF takes onBV(Ω) the general form

(1.1) F(u) =

Z

Ω

φ(|∇u|) dx+ Z

S_u

θ(|u⁺−u⁻|, ν_u) dHⁿ⁻¹+c₀|D^cu|(Ω),

where Du =∇uLⁿ+ (u⁺−u⁻)Hⁿ⁻¹+D^cuis the decomposition of the measure derivative of u in its absolutely continuous, jump and Cantor part, respectively, and S_u denotes the set of discontinuity points ofu.

The main difficulty in the actual minimization ofF comes from the surface integral Z

S_u

θ(|u⁺−u⁻|, νu) dHⁿ⁻¹,

which makes it necessary to use suitable approximations guaranteeing the convergence of minimum points and naturally leads to Γ-convergence.

As pointed out in [10], it is not possible to obtain a variational approximation for F by the typical integral functionals

Fε(u) = Z

Ω

fε(∇u) dx

defined on some Sobolev spaces. Indeed, when considering the lower semicontinuous envelopes of these functionals, we would be lead to a convex limit, which conflicts with the non-convexity of F.

Heuristic arguments suggest that, to get rid of the difficulty, we have to prevent that the effect oflargegradients is concentrated onsmall regions. Several approximation methods fit this requirements. For instance in [7], [12], [24] the case where the functionals Fε are restricted to finite elements spaces on regular triangulations of sizeεis considered. In [1], [2], [23] the implicit constraint on the gradient through the addition of a higher order penalization is investigated.

Moreover, it is important to mention theAmbrosio & Tortorelliapproximation (see [4] and [5]) of the Mumford-Shah functional via elliptic functionals.

The study of non-local models, where the effect of a large gradient is spread onto a set of size ε, was first introduced by Braides & Dal Maso in order to approximate the Mumford-Shah functional (see [10] and also [11], [13], [14], [15], [16]) by means of the family

(1.2) Fε(u) = 1

ε Z

Ω

f

ε Z

Bε(x)∩Ω

|∇u|²dy

dx, u∈H¹(Ω),

where, for instance,f(t) =t∧1/2 and B_ε(x) denotes the ball of centrexand radiusε. A variant of the method proposed in [10] has been used in [22] to deal with the approximation of a functional F of the form (1.1), with φhaving linear growth and θ independent on the normal ν_u (see also [20] and [21]). More precisely, in [22] the Γ-limit of the family

Fε(u) =1 ε

Z

Ω

f

ε Z

B_ε(x)∩Ω

|∇u|dy

dx, u∈W^1,1(Ω), for a suitable concave functionf, is computed.

In [25] (see also [13]) the case of an anisotropic variant of (1.2) has been considered. In particular it is proven that the family

Fε(u) = 1 ε Z

Ω

f ε|∇u|^p∗ρε

dx, u∈H¹(Ω), p >1, Γ-converges to an anisotropic version of the Mumford-Shah functional.

In this paper we investigate the Γ-convergence of the family Fε(u) = 1

ε Z

Ω

fε ε|∇u| ∗ρε

dx, u∈W^1,1(Ω).

The main difficulty to overcome is the estimate from below for the lower Γ-limit in terms of the surface part, while the contribution arising from the volume and Cantor parts has been treated along the same line of the argument already exploited in [25]. The estimate from above has

been achieved by density and relaxation arguments. We prove that the Γ-limit, in the strong L¹-topology, is given by

F(u) = Z

Ω

φ(|∇u|) dx+ Z

S_u

θ(|u⁺−u⁻|, ν_u) dHⁿ⁻¹+c₀|D^cu|(Ω), wherec₀= lim_t→+∞φ(t)/tand

θ(s, ν) = inf

lim inf

j→+∞

1 εj

Z

Q_ν

f(ε_j|∇u_j| ∗ρ_εj) dx: (u_j)∈W_ν^0,s, ε_j→0⁺

,

being W_ν^a,b the space of all sequences on the cylinder Q_ν which converge, shrinking onto the interface, to the function that jumps fromatobaround the origin (see paragraph 3.1 for details).

In section 7 we have been able to show that the method used in [22] to writeθ in a more explicit form works only if n = 1. In the case n > 1 such an argument does not work. Let us briefly discuss the reason. Without loss of generality we can supposeν =e1. LetP_C^⊥ be the orthogonal projection of Conto{x1 = 0}. Denote byX the space of all functions v∈W_loc^1,1(R×P_C^⊥) which are non-decreasing in the first variable and such that there existξ0< ξ1with v(x) = 0 ifx1< ξ0

andv(x) =sifx₁> ξ₁. Then, exploiting the same argument as in [22], we haveθ(s,e₁)≥inf_XG, where

G(v) = Z +∞

−∞

f Z

C(se1)

∂1v(z)ρ(z−te1) dz

dt.

The estimateθ(s,e1)≥infXGturns out to be optimal if infXG= infY G, whereY is the space of all functions v ∈ X such that v depends only on the first variable. This is due to the fact that proving the inequality θ(s,e1) ≥ infXG we lose control on all the derivatives ∂iv for any i = 2,· · ·, n. In the case C = B1 and ρ = _ω¹

n

χ_B

1, treated in [22], one is able to prove that infXG= infY G computing directly infXG by a discretization argument (see Prop. 5.7 in [22]).

In general, infXG= infYGdoes not hold. Indeed proceeding at first as in the proof of Prop. 5.6 in [22], one is able to show that for any C ⊂ R² open, bounded, convex and symmetrical set (i.e.C=−C) and forρ= _|C|¹ χ_C, it holds

(1.3) inf

Y G= Z h1

−h1

f s

|C|H¹(C∩ {z1=t}

dt.

Now ifC is the parallelogramC={(x, y)∈R²:−2≤y≤2, x−1≤y≤x+ 1}applying (1.3), we get

infY G⁰ = 2f 2s

|C|

+ 2

Z 2 0

f sr

|C|

dr.

If we computeGon the functionwgiven by w(x, y) =

0 ify > x−1 s ify≤x−1 ,

(to do this we notice that the functionalG makes sense also onBVloc(R×(−2,2)) writing D1v instead of∂1vdz) we obtain

G(w) = 2f 4s

|C|

.

Iff is strictly concave then G(w)<2f

2s

|C|

+ 2f

2s

|C|

<2f 2s

|C|

+ 2

Z 2 0

f sr

|C|

dr= inf

Y G.

By a density argument we deduce that infXG <infY G.

As a conclusion, it seems that for a generic anisotropic convolution kernelρεthe expression for θcan not be further simplified whenn >1.

2. Notation and preliminaries

We will denote byL^p(Ω) and byW^k,p(Ω), fork∈N,k≥1, and for 1≤p≤+∞, respectively the classical Lebesgue and Sobolev spaces on Ω. The Lebesgue measure of a measurable setA⊂Rⁿ will be denoted by|A|, whereas the Hausdorff measure ofAof dimensionm < nwill be denoted by H^m(A). The ball centered inxwith radius r will be denoted by Br(x), while Br stands for Br(0); moreover, we will use the notationSⁿ⁻¹for the boundary ofB1 inRⁿ. The volume of the unit ball inRⁿ will be denoted byωn, with the convention ω0= 1. FinallyA(Ω) denotes the set of all open subsets of Ω.

2.1. Functions of bounded variation. For a thorough treatment ofBV functions we refer the reader to [3]. Let Ω be an open subset of Rⁿ. We recall that the spaceBV(Ω) of real functions of bounded variation is the space of the functions u ∈ L¹(Ω) whose distributional derivative is representable by a measure in Ω,i.e.

Z

Ω

u∂ϕ

∂xi

dx=− Z

Ω

ϕdDiu, ∀ϕ∈C_c^∞(Ω),∀i= 1, . . . , n,

for someRⁿ-valued measureDu= (D₁u, . . . , D_nu) on Ω. We say thatuhasapproximate limitat x∈Ω if there existsz∈Rsuch that

lim

r→0⁺

Z

Br(x)

|u(y)−z|dy= 0.

The setS_u where this property fails is called approximate discontinuity setof u.The vectorz is uniquely determined for any point x∈Ω\S_u and is called theapproximate limit of uat xand denoted by ˜u(x).We say thatxis anapproximate jump pointof the functionu∈BV(Ω) if there exista, b∈Randν ∈Sⁿ⁻¹ such thata6=band

(2.1) lim

r→0⁺

Z

B⁺_r(x,ν)

|u(y)−a|dy= 0, lim

r→0⁺

Z

B⁻_r(x,ν)

|u(y)−b|dy= 0,

whereB_r⁺(x, ν) ={y∈Br(x) : hy−x, νi>0}andB⁻_r(x, ν) ={y∈Br(x) : hy−x, νi<0}.The set of approximate jump points ofuis denoted by Ju.The triplet (a, b, ν),which turns out to be uniquely determined up to a permutation ofaandb and a change of sign ofν,is usually denoted by (u⁺(x), u⁻(x), νu(x)). On Ω\Su we set u⁺ =u⁻ = ˜u.It turns out that for any u∈BV(Ω) the setSu is countably (n−1)-rectifiable andHⁿ⁻¹(Su\Ju) = 0.Moreover,

Du J_u= (u⁺−u⁻)ν_uHⁿ⁻¹ J_u

andν_u(x) gives the approximate normal direction toS_uforHⁿ⁻¹-a.e. x∈S_u.

For a function u∈ BV(Ω) let Du = D^au+D^su be the Lebesgue decomposition of Du into absolutely continuous and singular part. We denote by ∇u the density of D^au; the measures D^ju:=D^su J_uandD^cu:=D^su (Ω\S_u) are called thejump partand theCantor partof the derivative, respectively. It holdsDu=∇uLⁿ+ (u⁺−u⁻)ν_uHⁿ⁻¹ J_u+D^cu. Let us recall the following important compactness Theorem inBV (see Th. 3.23 and Prop. 3.21 in [3]):

Theorem 2.1. Let Ω be a bounded open subset of Rⁿ with Lipschitz boundary. Every sequence (u_h)inBV(Ω)which is bounded inBV(Ω)admits a subsequence converging inL¹(Ω)to a function u∈BV(Ω).

We say that a function u ∈ BV(Ω) is a special function of bounded variation, and we write u ∈ SBV(Ω), if |D^cu|(Ω) = 0. We say that a function u ∈ L¹(Ω) is a generalized function of bounded variation, and we write u∈GBV(Ω), ifu^T := (−T)∨u∧T belongs toBV(Ω) for every T ≥0. Ifu∈GBV(Ω), the function∇ugiven by

(2.2) ∇u=∇u^T a.e. on{|u| ≤T}

turns out to be well-defined. Moreover, the set functionT 7→S_uT is monotone increasing; therefore, if we set S_u = S

T >0J_uT , for Hⁿ⁻¹-a.e.x ∈ S_u we can consider the functions of T given by (u^T)⁻(x), (u^T)⁺(x),ν_uT(x). It turns out that

(2.3) u⁻(x) = lim

T→+∞(u^T)⁻(x), u⁺(x) = lim

T→+∞(u^T)⁺(x), ν_u(x) = lim

T→+∞ν_uT(x)

are well-defined for Hⁿ⁻¹-a.e.x ∈ S_u Finally, for a function u ∈ GBV(Ω), let |D^cu| be the supremum, in the sense of measures, of |D^cu^T| for T >0. It can be proved that for any Borel subsetB of Ω

(2.4) |D^cu|(B) = lim

T→+∞|D^cu^T|(B).

2.2. Slicing. In order to obtain the estimate from below of the lower Γ-limit (see next paragraph) we need some basic properties of one-dimensional sections of BV-functions. We first introduce some notation. Letξ ∈Sⁿ⁻¹, and let ξ^⊥ be the vector subspace orthogonal to ξ. If y ∈ξ^⊥ and E⊆Rⁿwe setE_ξ,y={t∈R: y+tξ∈E}.Moreover, for any given functionu: Ω→Rwe define u_ξ,y: Ω_ξ,y →Rbyu_ξ,y(t) =u(y+tξ). For the results collected in the following Theorem see [3], section 3.11.

Theorem 2.2. Let u∈ BV(Ω). Then u_ξ,y ∈ BV(Ω_ξ,y) for every ξ ∈ Sⁿ⁻¹ and for Hⁿ⁻¹-a.e.

y∈ξ^⊥. For such values ofy we haveu⁰_ξ,y(t) =h∇u(y+tξ), ξifor a.e.t∈Ωξ,y andJu_ξ,y = (Ju)ξ,y, whereu⁰_ξ,y denotes the absolutely continuous part of the measure derivative ofuξ,y. Moreover, for every open subsetA ofΩwe have

Z

ξ^⊥

|D^cuξ,y|(Aξ,y) dHⁿ⁻¹(y) =|hD^cu, ξi|(A).

2.3. Γ-convergence. For the general theory see [9] and [18]. Let (X, d) be a metric space. Let (F_j) be a sequence of functionsX →R. We say that (F_j) Γ-converges, asj→+∞, toF:X→R, if for allu∈X we have:

a) For every sequence (u_j) converging touit holds F(u)≤lim inf

j→+∞Fj(u_j).

b) There exists a sequence (u_j) converging to usuch that F(u)≥lim sup

j→+∞

Fj(uj).

ThelowerandupperΓ-limits of (Fj) inu∈X are defined as F⁰(u) = inf

lim inf

j→+∞Fj(uj) : uj→u , F⁰⁰(u) = inf lim sup

j→+∞

Fj(uj) : uj→u

respectively. We extend this definition of convergence to families depending on a real parameter.

Given a family (Fε)ε>0 of functionsX →R, we say that it Γ-converges, asε→0, toF:X →R if for every positive infinitesimal sequence (εj) the sequence (Fε_j) Γ-converges toF. If we define the lower and upper Γ-limits of (Fε) as

F⁰(u) = inf lim inf

ε→0 Fε(uε) : uε→u , F⁰⁰(u) = inf lim sup

ε→0

Fε(uε) : uε→u

respectively, then (Fε) Γ-converges to F in uif and only ifF⁰(u) =F⁰⁰(u) = F(u). It turns out that both F⁰ and F⁰⁰ are lower semicontinuous on X. In the estimate of F⁰ we shall use the following immediate consequence of the definition:

F⁰(u) = inf lim inf

j→+∞Fε_j(uj) : εj→0⁺, uj→u . It turns out that the infimum is attained.

An important consequence of the definition of Γ-convergence is the following result about the convergence of minimizers (see, e.g., [18], Cor. 7.20):

Theorem 2.3. LetFj:X→Rbe a sequence of functions whichΓ-converges to someF:X →R; assume thatinfv∈XFj(v)>−∞for everyj. Let(σj)be a positive infinitesimal sequence, and for every j letu_j ∈X be aσ_j-minimizer ofFj, i.e.

Fj(uj)≤ inf

v∈XFj(v) +σj.

Assume thatuj→ufor someu∈X. Thenuis a minimum point ofF, and F(u) = lim

j→+∞Fj(uj).

Remark 2.4. The following property is a direct consequence of the definition of Γ-convergence:

if Fε

→ FΓ then Fε+G→ F^Γ +G wheneverG:X →Ris continuous.

2.4. Supremum of measures. In order to prove the Γ-liminf inequality we recall the following useful tool, which can be found in [8].

Lemma 2.5. LetΩbe an open subset ofRⁿand denote byA(Ω)the family of its open subsets. Let λbe a positive Borel measure onΩ, andµ:A(Ω)→[0,+∞)a set function which is superadditive on open sets with disjoint compact closures, i.e. ifA, B⊂⊂ΩandA∩B=∅, then

µ(A∪B)≥µ(A) +µ(B).

Let (ψi)_i∈I be a family of positive Borel functions. Suppose that µ(A)≥

Z

A

ψidλ for every A∈ A(Ω) andi∈I.

Then

µ(A)≥ Z

A

sup

i

ψidλ for everyA∈ A(Ω).

2.5. A density result. The right bound for the upper Γ-limit from above will be first obtained for a suitable dense subset of SBV(Ω). More precisely, let W(Ω) be the space of all functions w∈SBV(Ω) such that

(a) Hⁿ⁻¹(S_w\S_w) = 0;

(b) Sw is the intersection of Ω with the union of a finite member of (n−1)-dimensional simplexes;

(c) w∈W^k,∞(Ω\Sw) for everyk∈N.

Theorem 3.1 in [17] gives us the density property ofW(Ω) we need; here SBV²(Ω) ={u∈SBV(Ω) :|∇u| ∈L²(Ω),Hⁿ⁻¹(S_u)<+∞}.

Theorem 2.6. Assume that ∂Ω is Lipschitz. Let u∈SBV²(Ω)∩L^∞(Ω). Then there exists a sequence (wh) in W(Ω) such thatwh →u strongly in L¹(Ω), ∇wh→ ∇ustrongly in L²(Ω,Rⁿ), withlim sup_h→+∞kwhk_∞≤ kuk_∞ and such that

lim sup

h→+∞

Z

S_wh

ψ(w⁺_h, w⁻_h, νw_h) dHⁿ⁻¹≤ Z

Su

ψ(u⁺, u⁻, νu) dHⁿ⁻¹

for every upper semicontinuous function ψ such that ψ(a, b, ν) = ψ(b, a,−ν) whenever a, b ∈ R andν ∈Sⁿ⁻¹.

2.6. A relaxation result. To conclude this section we prove a relaxation result which will be used in the sequel. Recall that givenX be a topological space andF:X →R∪ {±∞}, therelaxed functionalofF, denoted byF, is the largest lower semicontinuous functional which is smaller than F.

Theorem 2.7. Letφ: [0,+∞)→[0,+∞)be a convex, non-decreasing and lower semicontinuous function withφ(0) = 0 and with

t→+∞lim φ(t)

t =c∈(0,+∞).

Let θ: [0,+∞)×Sⁿ⁻¹ →[0,+∞)be a lower semicontinuous function such that θ(s, ν)≤c⁰s for any (s, ν)∈[0,+∞)×Sⁿ⁻¹, for somec⁰>0. For anyA∈ A(Ω)let

F(u, A) =









 Z

A

φ(|∇u|) dx+ Z

S_u∩A

θ(|u⁺−u⁻|, νu) dHⁿ⁻¹ ifu∈SBV²(Ω)∩L^∞(Ω)

+∞ otherwise inL¹(Ω).

Then the relaxed functional of F with respect to the strong L¹-topology satisfies F(u)≤

Z

Ω

φ(|∇u|) dx+ Z

S_u

θ(|u⁺−u⁻|, νu) dHⁿ⁻¹+c|D^cu|(Ω)

for any u∈BV(Ω).

Proof. Combining a standard convolution argument with a well known relaxation result (see, for instance, Th. 5.47 in [3]) we can say that the relaxed functional of

G(u, A) =









 Z

A

φ(|∇u|) dx ifu∈C¹(Ω) +∞ otherwise inL¹(Ω) is given by

G(u, A) =









 Z

A

φ(|∇u|) dx+c|D^su|(A) ifu∈BV(Ω)

+∞ otherwise inL¹(Ω).

SinceC¹(Ω)⊆SBV²(Ω)∩L^∞(Ω) then we get F(u, A)≤ G(u, A). Hence for anyA∈ A(Ω) and for anyu∈BV(Ω)

F(u, A)≤ Z

A

φ(|∇u|) dx+c|D^su|(A).

We can now conclude using the fact that for everyu∈BV(Ω) the set functionF(u,·) is the trace onA(Ω) of a regular Borel measureµ. This can be proven exactly along the same line of Prop. 3.3 in [6]. Hence

F(u) =µ(Ω) =µ(Ω\Su) +µ(Ω∩Su)

≤ Z

Ω

φ(|∇u|) dx+c|D^cu|(Ω) + Z

S_u

θ(|u⁺−u⁻|, νu) dHⁿ⁻¹

which is what we wanted to prove.

3. Statement of the main results

Let Ω⊂Rⁿ be a bounded open set with Lipschitz boundary. Letφ: [0,+∞)→[0,+∞) be a convex and non-decreasing function withφ(0) = 0 and

(3.1) lim

t→+∞

φ(t)

t =c0∈(0,+∞).

For anyε >0 letfε: [0,+∞)→[0,+∞) be such that:

A1) fε is non-decreasing, continuous, withfε(0) = 0.

A2) It holds lim

(ε,t)→(0,0)

fε(t) εφ _ε^t = 1.

A3) fε converges uniformly on the compact subsets of [0,+∞) to a concave functionf. Example 3.1. Given f andφas above, a possible choice for f_ε satisfying A1-A3 is given by

fε(t) =





 εφ _ε^t

if0≤t≤tε

f(t−tε) +εφ ^t_ε^ε

ift > tε

wheretε→0, andtε/ε→+∞. The only non-trivial assumption to verify is A2. Sinceε/tφ(t/ε)→ c0 as(ε, t)→(0,0), witht≥tε, the check amounts to verify that

lim

(ε,t)→(0,0) t≥tε

f(t−tε) +εφ ^t_ε^ε

t =c0.

This follows immediately fromf(t−tε)/(t−tε)→c0 andε/tεφ(tε/ε)→c0 as(ε, t)→(0,0), and t≥tε.

LetC⊂Rⁿbe open, bounded, and connected with 0∈C. Letρ:C→(0,+∞) be a continuous and bounded convolution kernel with

Z

C

ρdx= 1.

For anyε >0 and for anyx∈Rⁿ we will denote byCε(x) the setx+εC. For anyx∈εC let ρ_ε(x) = 1

εⁿρx ε

.

We consider the family (F_ε)_ε>0 of functionalsL¹(Ω)→[0,+∞] defined by

(3.2) Fε(u) =









 1 ε

Z

Ω

f_ε(ε|∇u| ∗ρ_ε) dx if u∈W^1,1(Ω)

+∞ otherwise inL¹(Ω)

where, for anyx∈Ω,

(3.3) |∇u| ∗ρε(x) =

Z

Cε(x)∩Ω

|∇u(y)|ρε(y−x) dy is a regularization by convolution of|∇u|by means of the kernel ρε. Remark 3.2. Notice that with the choiceC=B1 andρ= _ω¹

n

χ_B

1 we get

|∇u| ∗ρε(x) = Z

Bε(x)∩Ω

|∇u|dy

and thus the family(Fε)ε>0 reduces to the case already investigated in[20],[21] and[22].

In order to prove the Γ-convergence of Fε it is convenient to introduce a localized version of F_ε: more precisely, for eachA∈ A(Ω) we set

(3.4) Fε(u, A) =









 1 ε

Z

A

fε(ε|∇u| ∗ρε) dx if u∈W^1,1(Ω)

+∞ otherwise inL¹(Ω).

Clearly,Fε ·,Ω

coincides with the functionalFεdefined in (3.2). The lower and upper Γ-limits of Fε(·, A)

will be denoted byF⁰(·, A) andF⁰⁰(·, A), respectively.

3.1. The anisotropy. In this paragraph we define the surface density θ: [0,+∞)×Sⁿ⁻¹→[0,+∞) which will appear in the expression of the Γ-limit ofFε.

Givenν ∈Sⁿ⁻¹ anda, b∈Rlet us denote byu^a,b_ν the functionRⁿ→Rgiven by u^a,b_ν (x) =

a ifhx, νi<0 b ifhx, νi ≥0.

For anyx∈Rⁿ and anyν ∈Sⁿ⁻¹ letP_ν^⊥(x) be the orthogonal projection ofxonto the subspace ν^⊥ ={x∈Rⁿ :hx, νi= 0}. We define the cylinder

Qν ={x∈Rⁿ:|hx, νi| ≤1, P_ν^⊥(x)∈B1∩ν^⊥}.

Given Ω⁰ ⊂Rⁿ with Qν ⊂⊂Ω⁰ denote by W_ν^a,b the space of all sequences (uj) inW_loc^1,1(Ω⁰) such that uj →u^a,b_ν in L¹(Ω⁰), and such that there exist two positive infinitesimal sequences (aj),(bj) withuj(x) =aifhx, νi<−aj anduj=b ifhx, νi> bj. Let

(3.5) θ(s, ν) = 1 ω_n−1inf

lim inf

j→+∞

1 εj

Z

Qν

f(εj|∇uj| ∗ρε_j) dx: (uj)∈W_ν^0,s, εj →0⁺

. Notice thatθ(s, ν) does not depend on the choice of Ω⁰. Let us collect some easy properties ofθ which immediately descend from the definition.

Lemma 3.3. The following properties hold:

(3.6) θ is continuous.

(3.7) θ(s, ν) =θ(s,−ν), ∀s≥0, ∀ν ∈Sⁿ⁻¹.

(3.8)

inf

lim inf

j→+∞

1 εj

Z

Q_ν

f(ε_j|∇uj| ∗ρ_εj) dx: (u_j)∈W_ν^0,s, ε_j →0⁺

= inf

lim inf

j→+∞

1 εj

Z

Q_ν

f(εj|∇uj| ∗ρε_j) dx: (uj)∈W_ν^a,b, εj →0⁺

whenever|a−b|=s.

Moreover, for any x0∈Rⁿ,ν ∈Sⁿ⁻¹ ands≥0 we have (3.9) θ(s, ν) = 1

ω_n−1inf

lim inf

j→+∞

1 εj

Z

x₀+Q_ν

f(εj|∇uj| ∗ρε_j) dx: (uj(· −x0))∈W_ν^0,s, εj→0⁺

.

3.2. Main results. We are now in position to state the main result of the paper.

Theorem 3.4. Let F_εbe as in (3.2), withf_ε satisfying conditions A1-A3. ThenF_εΓ-converges, with respect to the strong L¹-topology, asε→0, toF: L¹(Ω)→[0,+∞]given by

F(u) =









 Z

Ω

φ(|∇u|) dx+ Z

S_u

θ(|u⁺−u⁻|, νu) dHⁿ⁻¹+c0|D^cu|(Ω) if u∈GBV(Ω)

+∞ otherwise in L¹(Ω).

Remark 3.5. Notice that for any u∈GBV(Ω) the expression θ(|u⁺−u⁻|, νu)turns out to be well definedHⁿ⁻¹-a.e.x∈Su, since (3.7)holds.

The proof of Theorem 3.4 will descend combining Proposition 5.10 (the Γ-liminf inequality) with Proposition 6.3 (the Γ-limsup inequality).

As a typical consequence of a Γ-convergence result, we are able to prove a result of convergence of minima by means of the following compactness result for equibounded (in energy) sequences, which will be proved in§4.

Theorem 3.6. Let (ε_j)be a positive infinitesimal sequence, and let (u_j)be a sequence inL¹(Ω) such that ||u_j||_∞ ≤M, and such thatF_εj(u_j)≤M for some positive constantM independent of j. Then the sequence(uj)converges, up to a subsequence, in L¹(Ω) to a function u∈BV(Ω).

Theorem 3.7. Let (εj) be a positive infinitesimal sequence and let g ∈L^∞(Ω). For everyu ∈ L¹(Ω) andj∈Nlet

Ij(u) =Fε_j(u) + Z

Ω

|u−g|dx, I(u) =F(u) + Z

Ω

|u−g|dx . For everyj letuj ∈L¹(Ω) be such that

Ij(uj)≤ inf

L¹(Ω)

Ij+εj.

Then the sequence(uj)converges, up to a subsequence, to a minimizer of I inL¹(Ω).

Proof. Since g ∈ L^∞(Ω) and since Fεj decreases by truncation, we can assume that (u_j) is equibounded in L^∞(Ω); for instance ||u_j||_∞ ≤ ||g||_∞. Applying Theorem 3.6 there exists u ∈ BV(Ω) such that (up to a subsequence)u_j→uinL¹(Ω). By Theorem 2.3, since (I_j) Γ-converges toI (see Th. 3.4 and Remark 2.4), uis a minimum point of I onL¹(Ω).

4. Compactness

In this section we prove Theorem 3.6. Let us first recall a useful technical Lemma which can be found in [10], Prop. 4.1. Actually such a Proposition has been proved for|∇u|², but, up to simple modifications, the same proof works for|∇u|.

For everyA∈ A(Ω) andσ >0 we set

A_σ={x∈A: d(x, ∂A)> σ}.

Lemma 4.1. Let g: [0,+∞)→[0,+∞)be a non-decreasing continuous function such that

t→0lim g(t)

t =c

for some c >0. LetA∈ A(Ω) withA⊂⊂Ω, and let u∈W^1,1(Ω)∩L^∞(Ω). For anyδ >0 and for any ε >0 sufficiently small, there exists a function v∈SBV(A)∩L^∞(A) such that

(1−δ) Z

A

|∇v|dx≤1 ε

Z

A

g

ε Z

B_ε(x)

|∇u|dy

dx,

Hⁿ⁻¹(Sv∩A6ε)≤c⁰ ε

Z

A

g

ε Z

Bε(x)

|∇u|dy

dx, kvk_L^∞_(A)≤ kuk_L^∞_(A)

kv−uk_L1(A_6ε)≤c⁰kuk_L∞(A)

Z

A

g

ε Z

Bε(x)

|∇u|dy

dx, wherec⁰ is a constant depending only onn, δ andg.

Proof of Theorem 3.6. LetA∈ A(Ω) withA⊂⊂Ω and∂Asmooth. Letr >0 such thatBr⊂C, and letm= infB_rρ >0. Then for anyx∈Awe haveBrε_j(x)⊂Cε_j(x) and thus forj sufficiently large,

|∇u_j| ∗ρ_εj(x) = Z

C_εj(x)

|∇u_j(y)|ρ_εj(y−x) dy≥ m εⁿ_j

Z

B_rεj(x)

|∇u_j(y)|dy

=mrⁿωn

Z

B_rεj(x)

|∇uj(y)|dy

for any x∈ A. Fixδ > 0. By A2 there exist t_δ >0 and j_δ such thatf_εj(t)≥(1−δ)ε_jφ(t/ε_j) for anyt ∈[0, t_δ] and j > j_δ. Letα, β ∈R, withα > 0 and β <0, be such that φ(t)≥αt+β everywhere. Then, sincef_εj is non-decreasing, we havef_εj(t)≥g_ε^δ

j(t) for anyt≥0, being g^δ_ε

j(t) =

(1−δ)αt+εjβ ift∈[0, tδ] (1−δ)αtδ+εjβ ift > tδ. Therefore, lettinghδ(t) =g_ε^δ_j(t)−εjβ, we have

(4.1)

Fεj(uj, A)≥ 1 εj

Z

A

hδ(|∇uj| ∗ρεj) dx+β|A|

≥ 1 εj

Z

A

hδ

mrⁿωnεj

Z

B_rεj(x)

|∇uj|dy

dx+β|A|.

Letηj =rεj andgδ,m,r(t) =¹_rgδ(mrⁿ⁻¹ωnt).Notice that, by construction,

t→0lim

gδ,m,r(t) t exists and is finite. Then inequality (4.1) becomes

Fε_j(uj, A)−β|A| ≥ 1 ηj

Z

Ω

gδ,r,m

ηj

Z

B_ηj(x)

|∇uj|dy

dx.

Applying Lemma 4.1 we find a sequence (v_j) inSBV(A) and a constantCindependent ofAsuch thatkvjkBV(A)≤C andkvjkL^∞(A)≤C.Moreover,

(4.2) kvj−ujk_L1(A)→0.

Hence, by Theorem 2.1, the sequence (vj) converges, up to a subsequence not relabeled, to some u ∈ BV(A), with kukBV(A) ≤ C. By (4.2) also u_j converges to u in L¹(A). The arbitrariness ofA and a diagonal argument allow to find a subsequence (u_jk) which converges in L¹_loc(Ω) to a functionu∈BV_loc(Ω), and the uniform bound of kujkL^∞(Ω)implies the convergence is strong in

L¹(Ω).

5. TheΓ-liminf inequality In this section we will prove that for anyu∈L¹(Ω) the inequality

F(u)≤lim inf

j→+∞Fε_j(uj)

holds for anyuj→uinL¹(Ω). First we will investigate two particular situations.

5.1. A preliminary estimate from below in terms of the volume and Cantor parts. In this paragraph we will take into account a simpler family of functionals. Let α, β > 0 and let g: [0,+∞)→[0,+∞) given byg(t) =αt∧β. LetG_ε:L¹(Ω)× A(Ω)→[0,+∞] be defined by

Gε(u, A) =









 1 ε

Z

A

g(ε|∇u| ∗ρ_ε) dx if u∈W^1,1(Ω)

+∞ otherwise inL¹(Ω).

We wish to estimate from below the lower Γ-limitG⁰(·, A) in terms of the volume and the Cantor parts ofDu. To this sake, we apply a slicing procedure, so that at first we will establish a suitable one-dimensional inequality. The idea of the proof is the same as in [25], where the superlinear growth case is treated.

Letm∈Nodd, letAbe an open interval inR, and let (εj) be a positive infinitesimal sequence.

LetAj={x∈εjZ:x∈A}.For anyj∈Nand for anyx∈Aj we define the interval I_j(x) =h

x−mε_j

2 , x+mε_j 2

i .

Lemma 5.1. Let α⁰, β⁰ > 0 and let hj: [0,+∞) → [0,+∞) given by hj(t) = α⁰t∧ ^β_ε⁰

j. Let u∈BV(A)and letu_j→uinL¹(A) withu_j∈W^1,1(A)for anyj∈N. Then

(5.1) lim inf

j→+∞ε_j X

x∈Aj

h_j Z

I_j(x)

|u⁰_j|dy

≥α⁰ Z

A

|u⁰|dy+α⁰|D^cu|(A).

Proof. For anyj∈Nandi= 0, . . . , m−1 letAⁱ_j = (iεj+mεjZ)∩A.ObviouslyAj is the disjoint union ofAⁱ_j fori∈ {0, . . . , m−1}. Then

X

x∈Aj

h_j Z

I_j(x)

|u⁰_j|dy

≥ 1 m

m−1

X

i=0

X

x∈Aⁱ_j

mh_j Z

I_j(x)

|u⁰_j|dy

.

Now let

Aⁱ_j = (

x∈Aⁱ_j: Z

I_j(x)

|u⁰_j|dx≤ β⁰ α⁰εj

)

and letv_j∈SBV(A) given by vj(x) =







u_j(x) ifx∈S

y∈Aⁱ_jI_j(y) 0 otherwise inA.

Hence X

x∈Aⁱ_j

mε_jh_j Z

Ij(x)

|u⁰_j|dy

≥ X

x∈Aⁱ_j

mε_jh_j Z

Ij(x)

|u⁰_j|dy

=α⁰ X

x∈Aⁱ_j

Z

Ij(x)

|u⁰_j|dy

=α⁰ Z

A

|v⁰_j|dy.

Observe that since we can suppose, without loss of generality, that ε_j X

x∈Aj

h_j Z

I_j(x)

|u⁰_j|dy

≤M

for someM ≥0, we deduce that M ≥εj

X

x∈Aj\Sm−1 i=0 Aⁱ_j

hj

Z

Ij(x)

|u⁰_j|dy

=εj

β⁰ εj

]

Aj\

m−1

[

i=0

Aⁱ_j

from which necessarily we have ε_j]

A_j\

m−1

[

i=0

Aⁱ_j

→0, asj →+∞.

This implies that ||uj−vj||L¹(A) →0 asj →+∞. Therefore, vj →uin L¹(A). Finally, by the superadditivity of the lim inf and by the lower semicontinuity of the total variation, we get

lim inf

j→+∞ε_j X

x∈Aj

h_j Z

I_j(x)

|u⁰_j|dy

≥ 1 m

m−1

X

i=0

lim inf

j→+∞

X

x∈Aⁱ_j

mε_jh_j Z

I_j(x)

|u⁰_j|dy

≥α⁰lim inf

j→+∞

Z

A

|v_j⁰|dy≥α⁰|Du|(A)

≥α⁰ Z

A

|u⁰|dy+α⁰|D^cu|(A)

which ends the proof.

Now, by applying the slicing Theorem 2.2, we will reduce the n-dimensional inequality to the one-dimensional inequality 5.1. Fix ξ∈Sⁿ⁻¹ and δ∈(0,1); consider an orthonormal basis {ei} withen=ξ. Let

Q^ξ_δ =

x∈Rⁿ:|hx,eii| ≤ δ

2, i= 1, . . . , n

, Q^ξ_δ(x) =x+Q^ξ_δ

and the lattice Z_δ^ξ = {x ∈ Rⁿ : hx,eii ∈ δZ, i = 1, . . . , n}. In what follows we will denote by gj(t) =_ε¹

jg(εjt); in particular it holds gj(t) =αt∧ _ε^β

j and G_εj(u, A) =

Z

A

g_j(|∇u| ∗ρ_εj) dx, u∈W^1,1(Ω).

Finally fix A∈ A(Ω) and letA^ξ_δ ={x∈ Z_δ^ξ : Q^ξ_δ(x) ⊂A}. The following Lemma is a standard easy application of the mean value Theorem (see also Lemma 4.2 in [10]).

Lemma 5.2. Let u∈W^1,1(Ω). Then there existsτ ∈Q^ξ_δ such that Gε_j(u, A)≥ X

x∈A^ξ_δ

δⁿgj(|∇u| ∗ρε_j(x+τ)).

Proof. We have

Gε_j(u, A)≥ X

x∈A^ξ_δ

Z

Q^ξ_δ(x)

gj(|∇u| ∗ρε_j(y)) dy= Z

Q^ξ_δ

X

x∈A^ξ_δ

gj(|∇u| ∗ρε_j(y+x)) dy.

Applying the mean value Theorem we get Z

Q^ξ_δ

X

x∈A^ξ_δ

gj(|∇u| ∗ρε_j(y+x)) dy= X

x∈A^ξ_δ

gj(|∇u| ∗ρε_j(τ+x))

for someτ ∈Q^ξ_δ, which concludes the proof.

We are in position to apply the slicing procedure.

Proposition 5.3. Let u∈BV(Ω) andA∈ A(Ω). Then G⁰(u, A)≥α

Z

A

|∇u|dx and G⁰(u, A)≥α|D^cu|(A).

Proof. Fixξ∈Sⁿ⁻¹. For anyη >0 letP_η^ξbe the union of the squaresQ^ξ_η(yi)⊂Cwithyi∈Z_η^ξ for i= 1, . . . , m, for somem∈Ndepending onη andξ. Let ρη be a non-negative constant function on the squaresQ^ξ_η(y_i) with 0< ρ_η≤ρand such that

c_η= Z

C

ρ_ηdx→1, as η→0.

Letc_i =ρ_η(y_i); then we can rewrite c_η as c_η =Pm

i=1c_iηⁿ.LetP_ηε^ξ

j be the union of the squares Q^ξ_ηεj(yi)⊆Cε_j, withyi ∈Z_ηε^ξ _j, fori= 1, . . . , m. LetA^ξ_j =A^ξ_ηεj; applying Lemma 5.2, since we can suppose, without loss of generality, thatuj∈W^1,1(Ω), there existsτj ∈Q^ξ_ηεj such that

Gε_j(uj, A)≥ X

x∈A^ξ_j

(ηεj)ⁿgj(|∇uj| ∗ρε_j(x+τj)).

LetB ⊂⊂A, and, for anyj sufficiently large, letv_j(y) =u_j(y+τ_j). Then we getv_j ∈W^1,1(B) andv_j →uin L¹(B). Thus

G_εj(u_j, A)≥ X

x∈B_j^ξ

(ηε_j)ⁿg(|∇v_j| ∗ρ_εj(x))

being B_j^ξ ={x∈Z_ηε^ξ _j : Q^ξ_ηεj ⊆B}. Now, for eachx∈B_j^ξ, we estimate the term|∇vj| ∗ρε_j(x);

we have, forj large enough,

|∇vj| ∗ρε_j(x) = Z

C_εj

|∇vj(y+x)|ρε_j(y) dy≥ 1 εⁿ_j

Z

P_ηεj^ξ

|∇vj(y+x)|ρη

y ε_j

dy

≥ 1 εⁿ_j

m

X

i=1

ci

Z

Q^ξ_ηεj(yi)

|∇vj(y+x)|dy=

m

X

i=1

ciηⁿ cη

Z

Q^ξ_ηεj(yi)

cη|∇vj(y+x)|dy.

SincePm i=1

ciηⁿ

cη = 1 and sinceg_j is concave we get, for everyx∈B_j^ξ, gj(|∇vj| ∗ρε_j(x))≥

m

X

i=1

ciηⁿ c_η gj

cη

Z

Q^ξ_ηεj(yi)

|∇vj(y+x)|dy

.

Thus, reordering the terms, we deduce that Gε_j(uj, A)≥ X

x∈D_j^ξ

(ηεj)ⁿgj

cη

Z

Q^ξ_ηεj(x)

|∇vj|dz

for any D ⊂⊂ B and j sufficiently large, being, as usual, D^ξ_j = {x ∈ Z_ηε^ξ _j : Q^ξ_ηεj ⊆ D}. For convenience we can suppose∇vj = 0 on

Rⁿ\ [

Q^ξ_ηεj⊆D

Q^ξ_ηεj.

Let hξi be the one-dimensional space generated by ξ. Let us denote by Zηε^ξkj and by Z_ηε^ξ⊥_j the orthogonal projections ofZ_ηε^ξ _j respectively on hξiand ξ^⊥. Then

G_εj(u_j, A)≥ X

x∈Z_ηεj^ξ

(ηε_j)ⁿg_j

c_η Z

Q^ξ_ηεj(x)

|∇v_j|dz

≥ X

x⊥∈Z^ξ_ηεj^⊥

X

x_k∈Z^ξ_ηεj^k

(ηεj)ⁿgj

cη

Z

Q^ξ_ηεj(x_⊥+x_k)

|∇vj|dz

wherex=x_k+x⊥ turns out to be the unique decomposition of anyx∈Z_ηε^ξ _j withx_k∈Zηε^ξkj and x_⊥ ∈Z_ηε^ξ⊥_j. Moreover, denoting by Q^ξηε^k_j and by Q^ξ_ηε^⊥_j the projections of Q^ξ_ηεj respectively on hξi and onξ^⊥, applying Jensen’s inequality we deduce that

Gε_j(uj, A)≥ X

x⊥∈Z_ηεj^ξ⊥

X

x_k∈Z_ηεj^ξk

(ηεj)ⁿgj

cη

Z

Q^ξ_ηεj^⊥(x_⊥)

Z

Q^ξ_ηεj^k (x_k)

|h∇vj(z_⊥+z_k), ξi|dz_kdz_⊥

≥ X

x⊥∈Z_ηεj^ξ⊥

X

x_k∈Z_ηεj^ξk

(ηεj)ⁿ Z

Q^ξ_ηεj^⊥(x⊥)

gj

cη

Z

Q^ξ_ηεj^k (xk)

|h∇vj(z_⊥+z_k), ξi|dz_k

dz_⊥

≥ X

x⊥∈Z_ηεj^ξ⊥

Z

Q^ξ_ηεj^⊥(x⊥)

X

x_k∈Z_ηεj^ξk

ηεjgj

cη

Z

Q^ξ_ηεj^k (xk)

|h∇vj(z_⊥+z_k), ξi|dz_k

dz_⊥

≥ Z

ξ^⊥

X

x_k∈Z_ηεj^ξk

ηεjgj

cη

Z

Q^ξ_ηεj^k (xk)

|h∇vj(z_⊥+z_k), ξi|dz_k

dz_⊥.

For anyσ >0 small letDσ ={x∈D:d(x, ∂D)> σ} andD^x_σ^⊥ ={x∈Dσ :x=x_⊥+x_kξ, x_k ∈ R}, for x_⊥ ∈ ξ^⊥. For j sufficiently large, v_j(x_⊥+·) ∈ W^1,1(D^x_σ^⊥). Furthermore, v_j → u in L¹(D^x_σ^⊥) for a.e.x⊥∈ξ^⊥. Lethj(t) =gj(cηt); then, by the very definition of g, it is easy to see thath_j(t) =αc_ηt∧_ε^β

j.We are in position to apply Lemma 5.1 with choice α⁰ =αc_η andβ⁰=β.

Thus

lim inf

j→+∞

X

xk∈Z^ξ_ηεj^k

ηε_jg_j

c_η Z

Q^ξ_ηεj^k (x_k)

|h∇v_j(z_⊥+z_k), ξi|dz_k

= lim inf

j→+∞

X

x_k∈Z^ξ_ηεj^k

ηε_jh_j Z

Q^ξ_ηεj^k (x_k)

|h∇vj(z_⊥+z_k), ξi|dz_k

≥αcη

Z

D_σ^z⊥

|h∇u(z_⊥+z_k), ξi|dz_k+αcη|hD^cu(z_⊥+·), ξi|(D^z_σ^⊥).

Taking into account Theorem 2.2 and Fatou’s Lemma we conclude that lim inf

j→+∞Gεj(u_j, A)≥c_ηα Z

D_σ

|h∇u(z), ξi|dz+c_ηα|hD^cu, ξi(Dσ).

Sincec_η →1 asη→0, letσ→0 andD%A. Then (5.2) G⁰(u, A)≥α

Z

A

|h∇u(z), ξi|dz and G⁰(u, A)≥α|hD^cu, ξi|(A)

for any ξ ∈ Sⁿ⁻¹. From the first inequality, using the superadditivity of G⁰ and Lemma 2.5 we easily deduce that

G⁰(u, A)≥α Z

A

|∇u|dz.