<span class="mathjax-formula">$\Gamma $</span>-limits of convolution functionals

DOI:10.1051/cocv/2012018 www.esaim-cocv.org

Γ -LIMITS OF CONVOLUTION FUNCTIONALS Luca Lussardi

and Annibale Magni

Abstract. We compute theΓ-limit of a sequence of non-local integral functionals depending 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.

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

Received December 24, 2011.

Published online January 23, 2013.

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 disconti- nuity set(according to a terminology introduced in [19]). This means that the variable functionuis required to be smooth outside a surfaceK, depending onu, and bothuandKenter the structure of the functional, which takes the form given by

F(u, K) =

Ω\Kφ(|∇u|) dx+

K∩Ωθ(|u⁺−u⁻|, ν_K) dHⁿ⁻¹,

being Ω an open subset of Rⁿ, K is a (n−1)-dimensional compact subset of Rⁿ, |u⁺−u⁻| the jump of uacrossK,ν_K the normal direction to K, whileφand θ given positive functions, whereasHⁿ⁻¹ denotes the (n−1)-dimensional Hausdorff measure.

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

F(u) =

Ωφ(|∇u|) dx+

Su

θ(|u⁺−u⁻|, ν_u) dHⁿ⁻¹+c₀|D^cu|(Ω), (1.1) whereDu=∇uLⁿ+ (u⁺−u⁻)Hⁿ⁻¹+D^cuis the decomposition of the measure derivative ofuin its absolutely continuous, jump and Cantor part, respectively, S_u denotes the set of discontinuity points ofu, and ν_u is a choice of the unit normal atS_u.

Keywords and phrases.Free discontinuities,Γ-convergence, anisotropy.

1 Dipartimento di Matematica e Fisica “N. Tartaglia”, Universit`a Cattolica del Sacro Cuore, via dei Musei 41, 25121 Brescia, Italy.l.lussardi@dmf.unicatt.it

2 Mathematisches Institut Abt. f¨ur Reine Mathematik, Albert-Ludwigs Universit¨at Freiburg, Eckerstrasse 1, 79104 Freiburg im Breisgau, Germany.annibale.magni@math.uni-freiburg.de

Article published by EDP Sciences c EDP Sciences, SMAI 2013

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

Su

θ(|u⁺−u⁻|, νu) dHⁿ⁻¹,

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

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

F_ε(u) =

Ωf_ε(∇u) dx

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

Heuristic arguments suggest that, to get rid of the difficulty, we have to prevent that the effect of large gradients is concentrated on small regions. Several approximation methods fit this requirements. For instance in [7,12,24] the case where the functionalsFε 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 the Ambrosio and Tortorelli approximation (see [3,4]) of the Mumford–Shah functional viaelliptic 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 and Dal Maso in order to approximate the Mumford–Shah functional (see [10] and also [11,13–16]) by means of the family

Fε(u) =1 ε

Ω

f

ε

Bε(x)∩Ω|∇u|²dy

dx, u∈H¹(Ω), (1.2)

where, for instance,f(t) =t∧1/2 andB_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 functionalF of the form (1.1), withφ having linear growth andθindependent on the normalν_u (see also [20,21]). More precisely, in [22] theΓ-limit of the family

F_ε(u) =1 ε

Ωf

ε

Bε(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

F_ε(u) = 1 ε

Ωf

ε|∇u|^p∗ρ_ε

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

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

ε

Ω

f_ε

ε|∇u| ∗ρ_ε

dx, u∈W^1,1(Ω),

where the family (f_ε)_ε>0 satisfies some conditions. The main difficulty to overcome is the estimate from below for the lowerΓ-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 theΓ-limit, in the strongL¹-topology, is given by

F(u) =

Ωφ(|∇u|) dx+

Su

θ(|u⁺−u⁻|, ν_u) dHⁿ⁻¹+c₀|D^cu|(Ω),

where φ(t) ∼ ¹_εf_ε(εt), as ε →0⁺, is a convex and non-decreasing function with φ(0) = 0 and with φ(t)/t → c₀>0 ast→+∞; moreover,

θ(s, ν) = inf

lim inf

j→+∞

1 ε_j

Qν

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

,

beingf the uniform limit, on compact subsets of [0,+∞), off_ε,W_ν^a,bthe space of all sequences on the cylinder Q_ν ={x∈Rⁿ :| x, ν| ≤1,the orthogonal projection ofxontoν^⊥ belongs to the unit ball},

which converge, shrinking onto the interface, to the function that jumps from a to b around the origin (see Sect.3.1for details).

In Section7we have been able to show that the method used in [22] to writeθin a more explicit form works only ifn= 1. In the casen >1 such an argument does not work. Let us briefly discuss the reason. Without loss of generality we can suppose ν =e₁. LetP_C^⊥ be the orthogonal projection of C onto{x₁ = 0}. Denote byX the space of all functionsv∈W_loc^1,1(R×P_C^⊥) which are non-decreasing in the first variable and such that there exist ξ₀ < ξ₁ with v(x) = 0 ifx₁< ξ₀and v(x) =sifx₁ > ξ₁. Then, exploiting the same argument as in [22], we haveθ(s,e₁)≥inf_XG, where

G(v) = _+∞

−∞ f

C(se1)

∂₁v(z)ρ(z−te₁) dz

dt.

The estimate θ(s,e₁) ≥ inf_XG turns out to be optimal if inf_XG = inf_Y G, where Y 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)≥inf_XGwe lose control on all the derivatives∂_iv for anyi= 2, . . . , n. In the caseC =B₁ and ρ = _ω¹

nχ_B

1, treated in [22], one is able to prove that inf_XG = inf_Y G computing directly inf_XG by a discretization argument (see Prop. 5.7 in [22]). In general, inf_XG= inf_Y Gdoes not hold. Indeed proceeding at first as in the proof of Proposition 5.6 in [22], one is able to show that for anyC⊂R² open, bounded, convex and symmetrical set (i.e.C=−C) and forρ= _|C|¹ χ_C, it holds

infY G= _h1

−h1

f s

|C|H¹(C∩ {z1=t}

dt. (1.3)

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

2s

|C|

+ 2 ₂

0

f sr

|C|

dr.

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

0 ify > x−1 sify≤x−1,

(to do this we notice that the functionalGmakes sense also onBV_loc(R×(−2,2)) writingD₁vinstead 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

₂

0

f sr

|C|

dr= inf

Y G.

By a density argument we deduce that inf_XG <inf_YG.

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

2. Notation and preliminaries

We will denote by L^p(Ω) and byW^k,p(Ω), fork∈N, k≥1, and for 1≤p≤+∞, respectively the classical Lebesgue and Sobolev spaces onΩ. The Lebesgue measure of a measurable set A⊂Rⁿ will be denoted by|A|, whereas the Hausdorff measure ofAof dimensionm < nwill be denoted byH^m(A). The ball centered inxwith radiusr will be denoted byB_r(x), whileB_r stands forB_r(0); moreover, we will use the notationSⁿ⁻¹ for the boundary ofB₁ inRⁿ. The volume of the unit ball inRⁿ will be denoted byω_n, with the conventionω₀= 1.

FinallyA(Ω) denotes the set of all open subsets ofΩ.

2.1. Functions of bounded variation

For a thorough treatment of BV functions we refer the reader to [5]. Let Ω be an open subset ofRⁿ. We recall that the space BV(Ω) of real functions of bounded variation is the space of the functions u ∈ L¹(Ω) whose distributional derivative is representable by a measure inΩ,i.e.

Ωu∂ϕ

∂x_idx=−

ΩϕdD_iu, ∀ϕ∈C_c^∞(Ω),∀i= 1, . . . , n,

for someRⁿ-valued measureDu= (D₁u, . . . , D_nu) onΩ. We say thatuhasapproximate limitatx∈Ωif there existsz∈Rsuch that

r→0lim⁺

Br(x)|u(y)−z|dy= 0.

The set S_u where this property fails is called approximate discontinuity set of u. The vector z is uniquely determined for any pointx∈Ω\S_uand is called theapproximate limitofuatxand denoted by ˜u(x).We say that xis anapproximate jump point of the functionu∈BV(Ω) if there exista, b∈Rand ν∈Sⁿ⁻¹ such that a=band

r→0lim⁺

Br⁺(x,ν)|u(y)−a|dy= 0, lim

r→0⁺

Br⁻(x,ν)|u(y)−b|dy= 0, (2.1) where B_r⁺(x, ν) = {y ∈ B_r(x) : y−x, ν > 0} and B_r⁻(x, ν) = {y ∈ B_r(x) : y−x, ν < 0}. The set of approximate jump points ofuis denoted byJ_u.The triplet (a, b, ν),which turns out to be uniquely determined up to a permutation ofaandband a change of sign ofν,is usually denoted by (u⁺(x), u⁻(x), ν_u(x)).OnΩ\S_u we set u⁺ = u⁻ = ˜u. It turns out that for anyu ∈ BV(Ω) the set S_u is countably (n−1)-rectifiable and Hⁿ⁻¹(S_u\J_u) = 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_u and D^cu:=D^su (Ω\S_u) are called the jump part and the Cantor part of the derivative, respectively. It holds Du =∇uLⁿ+ (u⁺−u⁻)ν_uHⁿ⁻¹ J_u+D^cu. Let us recall the following important compactness theorem in BV (see Thm. 3.23 and Prop. 3.21 in [5]):

Theorem 2.1. Let Ωbe a bounded open subset ofRⁿ 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 aspecial 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), if u^T := (−T)∨u∧T belongs toBV(Ω) for every T ≥0. If u∈ GBV(Ω), the function ∇u given by

∇u=∇u^T a.e. on{|u| ≤T} (2.2)

turns out to be well-defined. Moreover, the set functionT → S_uT is monotone increasing; therefore, if we set S_u= _{T >0}J_uT,forHⁿ⁻¹-a.e.x∈S_u we can consider the functions ofT given by (u^T)⁻(x), (u^T)⁺(x),ν_uT(x).

It turns out that

u⁻(x) = lim

T→+∞(u^T)⁻(x), u⁺(x) = lim

T→+∞(u^T)⁺(x), ν_u(x) = lim

T→+∞ν_uT(x) (2.3) are well-defined forHⁿ⁻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|forT >0. It can be proved that for any Borel subset B ofΩ

|D^cu|(B) = lim

T→+∞|D^cu^T|(B). (2.4)

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 ofBV-functions. We first introduce some notation. Letξ∈Sⁿ⁻¹, and let ξ^⊥ be the vector subspace orthogonal toξ. Ify∈ξ^⊥andE ⊆Rⁿ we setE_ξ,y={t∈R: y+tξ∈E}.Moreover, for any given functionu:Ω→Rwe defineu_ξ,y:Ω_ξ,y →Rbyu_ξ,y(t) =u(y+tξ).For the results collected in the following theorem see [5], 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 of y we have u_ξ,y(t) = ∇u(y+tξ), ξ for a.e.t∈Ω_ξ,y andJ_uξ,y = (J_u)_ξ,y, where u_ξ,y denotes the absolutely continuous part of the measure derivative ofu_ξ,y. Moreover, for every open subsetA of Ωwe have

ξ^⊥|D^cu_ξ,y|(Aξ,y) dHⁿ⁻¹(y) =| D^cu, ξ|(A).

2.3. Γ -convergence

For the general theory see [9,18]. Let (X, d) be a metric space. Let (F_j) be a sequence of functions X →R.

We say that (Fj)Γ-converges, as j→+∞, toF:X →R, if for allu∈X we have:

(a) for every sequence (u_j) converging to uit holds

F(u)≤lim inf

j→+∞Fj(u_j);

(b) there exists a sequence (u_j) converging tousuch that F(u)≥lim sup

j→+∞ Fj(u_j).

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

lim inf

j→+∞F_j(u_j) : u_j→u

, F(u) = inf

lim sup

j→+∞ F_j(u_j) : u_j →u

respectively. We extend this definition of convergence to families depending on a real parameter. Given a family (F_ε)_ε>0of functionsX →R, we say that itΓ-converges, asε→0, toF:X →Rif for every positive infinitesimal sequence (ε_j) the sequence (Fεj)Γ-converges toF. If we define the lower and upperΓ-limits of (Fε) as

F(u) = inf

lim inf

ε→0 Fε(u_ε) : u_ε→u

, F(u) = inf

lim sup

ε→0 Fε(u_ε) : u_ε→u

respectively, then (Fε)Γ-converges toF in uif and only if F(u) =F(u) =F(u).It turns out that both F andFare lower semicontinuous onX. In the estimate ofF we shall use the following immediate consequence of the definition:

F(u) = inf

lim inf

j→+∞F_εj(u_j) : ε_j →0⁺, u_j→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 (seee.g.[18], Cor. 7.20):

Theorem 2.3. Let Fj: X→Rbe a sequence of functions whichΓ-converges to someF: X→R; assume that inf_v∈XFj(v)>−∞ for every j. Let(σ_j) be a positive infinitesimal sequence, and for every j let u_j ∈X be a σ_j-minimizer ofFj,i.e.

Fj(u_j)≤ inf

v∈XFj(v) +σ_j.

Assume thatu_j →ufor some u∈X. Then uis a minimum point ofF, and F(u) = lim

j→+∞F_j(u_j).

Remark 2.4. The following property is a direct consequence of the definition of Γ-convergence: if F_ε → F^Γ thenFε+G→ F^Γ +G wheneverG:X →Ris continuous.

2.4. Supremum of measures

In order to prove theΓ-liminf inequality we recall the following useful tool, which can be found in [8].

Lemma 2.5. Let Ω be an open subset of Rⁿ and denote by A(Ω) the family of its open subsets. Let λ be a positive Borel measure on Ω, and μ: A(Ω)→[0,+∞)a set function which is superadditive on open sets with disjoint compact closures,i.e. if A, B⊂⊂Ω andA∩B =∅, then

μ(A∪B)≥μ(A) +μ(B).

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

Aψ_idλ for every A∈ A(Ω)andi∈I.

Then

μ(A)≥

Asup

i ψ_idλ for every A∈ A(Ω).

2.5. A density result

The right bound for the upperΓ-limit from above will be first obtained for a suitable dense subset ofSBV(Ω).

More precisely, letW(Ω) be the space of all functions w∈SBV(Ω) such that (a) Hⁿ⁻¹(S_w\S_w) = 0;

(b) S_wis the intersection ofΩ with the union of a finite member of (n−1)-dimensional simplexes;

(c) w∈W^k,∞(Ω\S_w) 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 (w_h) inW(Ω) such thatw_h→ustrongly inL¹(Ω),∇w_h→ ∇ustrongly inL²(Ω,Rⁿ), withlim sup_h→+∞w_h∞≤ u∞ and such that

lim sup

h→+∞

S_whψ(w_h⁺, w⁻_h, ν_wh) dHⁿ⁻¹≤

Su

ψ(u⁺, u⁻, ν_u) dHⁿ⁻¹

for every upper semicontinuous functionψ such that ψ(a, b, ν) =ψ(b, a,−ν)whenevera, 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 and F:X →R∪ {±∞}, the relaxed functional of F, denoted byF, is the largest lower semicontinuous functional which is smaller thanF.

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

t→+∞lim φ(t)

t =c∈(0,+∞).

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

F(u, A) =

⎧⎪

⎨

⎪⎩

Aφ(|∇u|) dx+

Su∩Aθ(|u⁺−u⁻|, ν_u) dHⁿ⁻¹if u∈SBV²(Ω)∩L^∞(Ω)

+∞ otherwise in L¹(Ω).

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

Ωφ(|∇u|) dx+

Su

θ(|u⁺−u⁻|, ν_u) 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, Thm. 5.47 in [5]) we can say that the relaxed functional of

G(u, A) =

⎧⎪

⎨

⎪⎩

Aφ(|∇u|) dxifu∈C¹(Ω) +∞ otherwise inL¹(Ω) is given by

G(u, A) =

⎧⎪

⎨

⎪⎩

Aφ(|∇u|) dx+c|D^su|(A) ifu∈BV(Ω)

+∞ otherwise inL¹(Ω).

Since C¹(Ω) ⊆ SBV²(Ω)∩L^∞(Ω) then we get F(u, A) ≤ G(u, A). Hence for any A ∈ A(Ω) and for any u∈BV(Ω)

F(u, A)≤

Aφ(|∇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 Proposition 3.3 in [6]. Hence

F(u) =μ(Ω) =μ(Ω\S_u) +μ(Ω∩S_u)≤

Ωφ(|∇u|) dx+c|D^cu|(Ω) +

Su

θ(|u⁺−u⁻|, ν_u) 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. Let φ: [0,+∞)→ [0,+∞) be a convex and non-decreasing function withφ(0) = 0 and

t→+∞lim φ(t)

t =c₀∈(0,+∞). (3.1)

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

(A1) f_ε is non-decreasing, continuous, withf_ε(0) = 0.

(A2) It holds lim

(ε,t)→(0,0)

f_ε(t) εφ_t

ε

= 1.

(A3) f_ε converges uniformly on the compact subsets of [0,+∞) to a concave functionf. Example 3.1. Given f and φas above, a possible choice forf_εsatisfying A1–A3 is given by

f_ε(t) = εφ_t

ε

if 0≤t≤t_ε f(t−t_ε) +εφ_t

εε

ift > t_ε

where t_ε → 0, and t_ε/ε → +∞. The only non-trivial assumption to verify is A2. Since ε/tφ(t/ε) → c₀ as (ε, t)→(0,0), witht≥t_ε, the check amounts to verify that

(ε,t)→(0,0)lim

t≥tε

f(t−t_ε) +εφ_t

εε

t =c₀.

This follows immediately fromf(t−t_ε)/(t−t_ε)→c₀ andε/t_εφ(t_ε/ε)→c₀ as (ε, t)→(0,0), and t≥t_ε. LetC⊂Rⁿbe open, bounded, and connected with 0∈C. Letρ:C→(0,+∞) be a continuous and bounded

convolution kernel with

Cρdx= 1.

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

εⁿρ x

ε ·

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

F_ε(u) =

⎧⎪

⎨

⎪⎩ 1 ε

Ωf_ε(ε|∇u| ∗ρ_ε) dx if u∈W^1,1(Ω)

+∞ otherwise inL¹(Ω)

(3.2)

where, for anyx∈Ω,

|∇u| ∗ρ_ε(x) =

Cε(x)∩Ω|∇u(y)|ρε(y−x) dy (3.3)

is a regularization by convolution of|∇u| by means of the kernelρ_ε. Remark 3.2. Notice that with the choiceC=B₁ andρ=_ω¹

nχ_B

1 we get

|∇u| ∗ρ_ε(x) =

Bε(x)∩Ω|∇u|dy

and thus the family (F_ε)_ε>0 reduces to the case already investigated in [20–22].

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

Fε(u, A) =

⎧⎪

⎨

⎪⎩ 1 ε

Af_ε(ε|∇u| ∗ρ_ε) dx if u∈W^1,1(Ω)

+∞ otherwise inL¹(Ω).

(3.4)

Clearly,F_ε

·, Ω

coincides with the functional F_ε defined in (3.2). The lower and upper Γ-limits of

F_ε(·, A) will be denoted byF(·, A) andF(·, 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 ofF_ε.

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

a if x, ν<0 b if x, ν ≥0.

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

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

GivenΩ ⊂Rⁿ withQ_ν ⊂⊂Ω denote byW_ν^a,b the space of all sequences (u_j) inW_loc^1,1(Ω) such thatu_j→u^a,b_ν inL¹(Ω), and such that there exist two positive infinitesimal sequences (a_j),(b_j) withu_j(x) =aif x, ν<−aj

andu_j=bif x, ν> b_j. Let θ(s, ν) = 1

ω_n−1inf

lim inf

j→+∞

1 ε_j

Qν

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

. (3.5)

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

Lemma 3.3. The following properties hold:

θ is continuous. (3.6)

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

lim inf

j→+∞

1 ε_j

Qν

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

= inf

lim inf

j→+∞

1 ε_j

Qν

f(ε_j|∇u_j| ∗ρ_εj) dx: (u_j)∈W_ν^a,b, ε_j →0⁺ whenever|a−b|=s.

(3.8)

Moreover, for anyx₀∈Rⁿ,ν ∈Sⁿ⁻¹ ands≥0 we have θ(s, ν) = 1

ω_n−1inf

lim inf

j→+∞

1 ε_j

x0+Qν

f(ε_j|∇u_j| ∗ρ_εj) dx: (u_j(· −x₀))∈W_ν^0,s, ε_j→0⁺

. (3.9)

3.2. Main results

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

Theorem 3.4. Let F_ε be as in(3.2), withf_εsatisfying conditions A1–A3. Then F_ε Γ-converges, with respect to the strong L¹-topology, asε→0, toF: L¹(Ω)→[0,+∞] given by

F(u) =

⎧⎨

⎩

Ωφ(|∇u|) dx+

Su

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

+∞ otherwise in L¹(Ω).

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

The proof of Theorem 3.4 will descend combining Proposition 5.10 (the Γ-liminf inequality) with Proposition6.3(theΓ-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 Section4.

Theorem 3.6. Let (ε_j) be a positive infinitesimal sequence, and let (u_j) be a sequence in L¹(Ω) such that

||uj||∞ ≤M, and such that Fεj(u_j)≤M for some positive constant M independent of j. Then the sequence (u_j)converges, up to a subsequence, in L¹(Ω)to a function u∈BV(Ω).

Theorem 3.7. Let(ε_j)be a positive infinitesimal sequence and letg∈L^∞(Ω). For everyu∈L¹(Ω)andj∈N let

Ij(u) =Fεj(u) +

Ω|u−g|dx, I(u) =F(u) +

Ω|u−g|dx.

For every j letu_j ∈L¹(Ω)be such that

Ij(u_j)≤ inf

L¹(Ω)Ij+ε_j.

Then the sequence (u_j)converges, up to a subsequence, to a minimizer of I in L¹(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 ||uj||∞ ≤ ||g||∞. Applying Theorem 3.6 there exists u ∈ BV(Ω) such that (up to a subsequence)u_j→uinL¹(Ω). By Theorem2.3, since (Ij)Γ-converges toI (see Thm.3.4and Rem.2.4),uis

a minimum point ofI onL¹(Ω).

4. Compactness

In this section we prove Theorem3.6. Let us first recall a useful technical Lemma which can be found in [10], Proposition 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 somec >0. Let A∈ A(Ω) withA⊂⊂Ω, and letu∈W^1,1(Ω)∩L^∞(Ω). For any δ >0 and for any ε >0 sufficiently small, there exists a function v∈SBV(A)∩L^∞(A) such that

(1−δ)

A|∇v|dx≤ 1 ε

Ag

ε

Bε(x)|∇u|dy

dx, Hⁿ⁻¹(S_v∩A_6ε)≤c

ε

Ag

ε

Bε(x)|∇u|dy

dx, vL^∞(A)≤ uL^∞(A)

v−u_L¹_(A6ε)≤cu_L^∞_(A)

Ag

ε

Bε(x)|∇u|dy

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

Proof of Theorem 3.6. Let A ∈ A(Ω) with A ⊂⊂ Ω and ∂A smooth. Let r > 0 such that B_r ⊂ C, and let m= inf_Brρ >0. Then for anyx∈Awe haveB_rεj(x)⊂C_εj(x) and thus forj sufficiently large,

|∇uj| ∗ρ_εj(x) =

C_εj(x)|∇uj(y)|ρεj(y−x) dy ≥ m εⁿ_j

B_rεj(x)|∇uj(y)|dy=mrⁿω_n

B_rεj(x)|∇uj(y)|dy for anyx∈A. Fix δ >0. By A2 there existt_δ>0 andj_δ such that f_εj(t)≥(1−δ)ε_jφ(t/ε_j) for anyt∈[0, t_δ] and j > j_δ. Let α, β ∈R, with α >0 and β <0, be such that φ(t)≥αt+β everywhere. Then, since f_εj is non-decreasing, we havef_εj(t)≥g^δ_εj(t) for anyt≥0, being

g_ε^δ_j(t) =

(1−δ)αt+ε_jβ ift∈[0, t_δ] (1−δ)αt_δ+ε_jβ ift > t_δ. Therefore, lettingh_δ(t) =g^δ_εj(t)−ε_jβ, we have

F_εj(u_j, A)≥ 1 ε_j

Ah_δ(|∇u_j| ∗ρ_εj) dx+β|A| ≥ 1 ε_j

Ah_δ

mrⁿω_nε_j

B_rεj(x)|∇u_j|dy

dx+β|A|. (4.1)

Letη_j =rε_j andg_δ,m,r(t) =¹_rg_δ(mrⁿ⁻¹ω_nt).Notice that, by construction,

t→0lim

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

Fεj(u_j, A)−β|A| ≥ 1 η_j

Ωg_δ,r,m

η_j

B_ηj(x)|∇u_j|dy

dx.

Applying Lemma 4.1 we find a sequence (v_j) in SBV(A) and a constant C independent of A such that v_jBV(A)≤C andv_jL^∞_(A)≤C.Moreover,

v_j−u_jL¹_(A)→0. (4.2)

Hence, by Theorem2.1, the sequence (v_j) converges, up to a subsequence not relabeled, to someu∈BV(A), withu_BV(A)≤C.By (4.2) alsou_j converges touin L¹(A). The arbitrariness ofAand a diagonal argument allow to find a subsequence (u_jk) which converges in L¹_loc(Ω) to a function u ∈ BV_loc(Ω), and the uniform

bound ofuj_L^∞_(Ω)implies the convergence is strong inL¹(Ω).

5. The Γ -liminf inequality

In this section we will prove that for anyu∈L¹(Ω) the inequality F(u)≤lim inf

j→+∞Fεj(u_j)

holds for anyu_j→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. Letα, β >0 and letg: [0,+∞)→ [0,+∞) given by g(t) =αt∧β. LetGε:L¹(Ω)× A(Ω)→[0,+∞] be defined by

Gε(u, A) =

⎧⎪

⎨

⎪⎩ 1 ε

Ag(ε|∇u| ∗ρ_ε) dx if u∈W^1,1(Ω)

+∞ otherwise inL¹(Ω).

We wish to estimate from below the lowerΓ-limitG(·, 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.

Let m ∈ N odd, let A be an open interval in R, and let (ε_j) be a positive infinitesimal sequence. Let A_j ={x∈ε_jZ:x∈A}.For anyj∈Nand for anyx∈A_j we define the interval

I_j(x) =

x−mε_j

2 , x+mε_j 2

·

Lemma 5.1. Let α, β >0 and leth_j: [0,+∞)→[0,+∞) given byh_j(t) =αt∧^β_εj. Let u∈BV(A) and let u_j→uinL¹(A)with u_j∈W^1,1(A) for anyj∈N. Then

lim inf

j→+∞ε_j

x∈Aj

h_j

Ij(x)|u_j|dy

≥α

A|u|dy+α|D^cu|(A). (5.1) Proof. For anyj∈Nandi= 0, . . . , m−1 letAⁱ_j= (iε_j+mε_jZ)∩A.ObviouslyA_j is the disjoint union ofAⁱ_j fori∈ {0, . . . , m−1}. Then

x∈Aj

h_j

Ij(x)|u_j|dy

≥ 1 m

m−1

i=0

x∈Aⁱ_j

mh_j

Ij(x)|u_j|dy

.

Now let

Aⁱ_j=

x∈Aⁱ_j:

Ij(x)|u_j|dx≤ β αε_j

and letv_j∈SBV(A) given by

v_j(x) =

u_j(x) ifx∈ _y∈Ai jI_j(y) 0 otherwise in A.

Hence

x∈Aⁱ_j

mε_jh_j

Ij(x)|u_j|dy

≥

x∈Aⁱ_j

mε_jh_j

Ij(x)|u_j|dy

=α

x∈Aⁱ_j

Ij(x)|u_j|dy=α

A|v_j|dy.