A relaxation result for energies defined on pairs set-function and applications

Vol. 13, N^o 4, 2007, pp. 717–734 www.esaim-cocv.org

DOI: 10.1051/cocv:2007032

A RELAXATION RESULT FOR ENERGIES DEFINED ON PAIRS SET-FUNCTION AND APPLICATIONS

Andrea Braides

, Antonin Chambolle

and Margherita Solci

Abstract. We consider, in an open subset Ω ofR^N, energies depending on the perimeter of a subset E⊂Ω (or some equivalent surface integral) and on a functionuwhich is defined only on Ω\E. We compute the lower semicontinuous envelope of such energies. This relaxation has to take into account the fact that in the limit, the “holes”E may collapse into a discontinuity ofu, whose surface will be counted twice in the relaxed energy. We discuss some situations where such energies appear, and give, as an application, a new proof of convergence for an extension of Ambrosio-Tortorelli’s approximation to the Mumford-Shah functional.

Mathematics Subject Classification. 49J45, 49Q20, 49Q10.

Received December 12, 2005. Revised July 17, 2006.

Published online July 20, 2007.

1. Introduction

In this paper we consider energies defined on pairs function/open subset ofRⁿ, of the form F(u, E) =

Ω\E

f(∇u) dx+

∂Eϕ(ν) dHⁿ⁻¹,

with interacting bulk and surface energies. Here E is thought to be smooth enough (e.g., with Lipschitz boundary) so that ∂E coincides Hⁿ⁻¹-a.e. with the essential boundary of E; i.e., with the interface between the ‘interior’ and ‘exterior’ of E, and u∈ W^1,p(Ω;R^m). Energies of this type arise in physical problems for example when dealing with small drops or thin films, when bulk and surface energies can be thought to be of the same order (see, e.g., [7] for a variational problem set in this framework).

Functionals as F appear also in some disguised form in many problems related to variational models in Image Segmentation, such as that by Mumford and Shah [22, 23]. A particularly successful approach to deal with such problems has proven to be the application of the direct methods of the Calculus of Variations in the framework of the special functions of bounded variation to obtain existence and regularity results. In order to apply these existence results to Image Segmentation problems a further step is necessary, of approximating free-discontinuity energies, containing competing bulk and surface integrals, by energies to which numerical

Keywords and phrases. Relaxation, free discontinuity problems, Γ-convergence.

1 Dip. di Matematica, Universit`a di Roma “Tor Vergata”, via della Ricerca Scientifica, 00133 Roma, Italy.

2 CMAP, ´Ecole Polytechnique, CNRS, 91128 Palaiseau, France;antonin.chambolle@polytechnique.fr

3 DAP, Universit`a di Sassari, Palazzo Pou Salit, 07041 Alghero, Italy.

c EDP Sciences, SMAI 2007 Article published by EDP Sciences and available at http://www.esaim-cocv.org or http://dx.doi.org/10.1051/cocv:2007032

methods can be more easily applied. This has been done in many different ways, using elliptic energies with an additional variable, finite-difference energies, non-local integrals, etc. (see [11]). A commom pattern can be traced in all those approximations, that start from anansatzon the desired form of an ‘approximate solution’.

While it is easily seen, by construction of the approximating energies, that the candidate approximate solutions give the desired limit energy, more elaborate arguments are usually necessary to prove that the envisaged form of the solutions is ‘optimal’; i.e., that substantially lower energies cannot be achieved using a different type of pattern. To prove this fact (in the language of Γ-convergence we would call this the ‘liminf inequality’) a crucial point is, given an arbitrary sequence of minimizers, to distinguish sets in which the approximating energies computed on these functions behave as ‘bulk energies’, and complementary sets which we may regard as ‘blurred’ discontinuity sets (typically these are sets where ‘gradients are high’). This point can be rephrased as comparing the candidate approximating energies with an energy as F above defined on pairs function-set, whose form is in general dependent only on the ‘target’ free-discontinuity energy. In the case of the Mumford and Shah functional

MS(u) =

Ω

|∇u|²dx+Hⁿ⁻¹(S(u)∩Ω)

(S(u) denotes the set of discontinuity points of u), in this process of proving the liminf inequality we often produce energies of the form

F(u, E) =a

Ω\E|∇u|²dx+bHⁿ⁻¹(Ω∩∂E),

withaandbconstants belonging to some sets that depend on the specific argument used. The ‘liminf inequality’

is then rephrased in terms of the lower semicontinuous envelope ofF, and a localization argument showing that we may separately optimize the choice of a and b. This remark is essentially already contained in a paper by Bourdin and Chambolle [10], but therein it is not stated explicitly in terms of a relaxation result, and is obtained by applying more elaborated approximation results.

Note that at fixedEthe functionalF(·, E) is weakly lower semicontinuous onW^1,p, provided some standard convexity and growth conditions on f are required, and that, at fixed u, F(u,·) can be extended to a lower semicontinuous energy on sets with finite perimeter if ϕ is a norm (see for instance [3]). On the contrary, the functional defined on pairs (u, E) is not lower semicontinuous. Loosely speaking, if (u_j, E_j) is a sequence with equi-bounded energy and converging to some (u, E), the limit umay be discontinuous on Ω\E, and its set of discontinuity points S(u) may be the limit of a portion of∂E_j. In this paper we compute the lower- semicontinuous envelope of functionals F in a direct way, and characterize it in the whole class of pairs (u, E), whereEis a set of finite perimeter anduis such thatu(1−χ_E) belongs to the spaceGSBV(Ω;R^m) of Ambrosio and De Giorgi’s generalized special function of bounded variation (Th. 2). We show that it takes the form

F(u, E) =

Ω\E

f(∇u) dx+

∂^∗E∩Ωϕ(ν) dHⁿ⁻¹+

S(u)∩Ω∩E0

ϕ(ν) dHⁿ⁻¹,

whereE₀ denotes the points of density zero ofE andϕ(ν) = ϕ(ν) +ϕ(−ν). The result is proved by providing separately a lower bound and an upper bound. The lower bound is obtained by reducing to the one dimensional case through Ambrosio’s slicing techniques. The crucial technical point is here to check that, loosely speaking, for almost all directions, the traces of one-dimensional sections of E have density 0 on the jump set S(u) precisely on the intersection of E₀ and S(u) (Lem. 5). The upper bound is obtained by a direct construction if S(u) is smooth enough, and by approximation in the general case. We show two ways to obtain such an approximation. The first one (Lem. 12) consists in applying a ‘strong SBV approximation’ result by Braides and Chiad`o Piat [14] to a suitable modification of the function u(1−χ_E), and then construct optimal pair from this sequence of approximating functions. The second one (Rem. 13) uses a mollification argument for approximating the setEfirst, and then the coarea formula to select approximating sets, on which then to obtain an approximation of the targetu.

As applications of this result, we first give an approximation of the Mumford-Shah energy by a sequence of functions defined on pairs set-functions, by noting thatF(u, E) reduces to a functional onGSBV(Ω) when E=∅, so thatE₀= Ω. Subsequently, we give a different proof of Ambrosio-Tortorelli’s elliptic approximation result [4]. At the same time we provide a generalization by replacing their one-well potential by perturbed double-well potentials, which give a different smoother form of the optimal profile of solutions. This result formalizes a method that has already been used by Braides and March [15] to obtain minimizing sequences bounded inH²for problems in which a term penalizing the curvature of the discontinuity set is added. Finally, we outline applications to the study of crystalline films on a substrate and to water waves. As an interesting additional object for perspective work, we mention the interaction with boundary conditions and microgeometry, that would lead to interesting problems of homogenization, as shown by Alberti and De Simone [1] already in the case when no bulk term is present (see also an earlier paper by Bouchitt´e and Seppecher [8]). As a final bibliographical information, the results in this paper concerning the Ambrosio-Tortorelli approximation previously circulated in the form of the manuscript [16].

2. The relaxation result

Letf :R^m×n→Rbe a quasiconvex function satisfying the growth condition c₁(|ξ|^p−1)≤f(ξ)≤c₂(1 +|ξ|^p)

for some positive constants c₁ and c₂, and p > 1, and let ϕ : Rⁿ → [0,+∞) be a convex and positively homogeneous function of degree one, with ϕ(z)>0 if z = 0. For everyu∈ L¹(Ω) and E measurable subset of Ω,we define

F(u, E) =

⎧⎨

⎩

Ω\E

f(∇u) dx+

Ω∩∂E

ϕ(ν_E) dHⁿ⁻¹ if u∈W^1,p(Ω;R^m) and∂E Lipschitz

+∞ otherwise,

whereν_E denotes the interior normal toE.

We will prove the following relaxation theorem, in whose statement we adopt standard notation for generalized functions of bounded variations (see [5]); in particular, S(u) andν_u denote the set of essential discontinuity points ofuand its measure-theoretical normal, respectively. We say thatE_j→E ifχ_Ej →χ_EinL¹(Ω); M(Ω) denotes the family of measurable subsets of Ω and GSBV(Ω;R^m) the space of R^m-valued generalized special functions of bounded variation on Ω.

Theorem 1. The lower-semicontinuous envelope of the functionalF, with respect to theL¹(Ω;R^m)×L¹(Ω;R^m) topology, is the functionalF: L¹(Ω;R^m)× M(Ω)→[0,+∞]defined as:

F(u, E) =

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

Ω\E

f(∇u)dx+

Ω∩∂^∗E

ϕ(ν_E) dHⁿ⁻¹+

Ω∩S(u)∩E0

(ϕ(ν_u) +ϕ(−ν_u)) dHⁿ⁻¹ if uχ_E0∈GSBV(Ω;R^m)

+∞ otherwise,

where∂^∗E is the reduced boundary of E andE₀ is the set of the points whereE has density0.

Furthermore, if 0<|E| ≤ |Ω| then for every pair (u, E) there exists a recovery sequence(u_j, E_j)such that lim_jF(u_j, E_j) =F(u, E) and|E_j|=|E|.

The proof of Theorem 1 will be given in Sections 3 and 4, by showing, respectively, a lower and an upper inequality. By the nature of the problem we follow a ‘hybrid’ approach to the liminf inequality. In fact, on one hand we may regard the functionalF as the restriction of a functionalGdefined onGSBV(Ω;R^m) to functions v=uχ_Ω\E. A simple lower-semicontinuity argument applied to this functionalGgives the optimal lower bound

for the bulk term ofF (Rem. 7). This argument is not optimal for the surface energies inF, for which we follow a different path by using a slicing argument. Since the argument for this part is essentially scalar and does not involve the bulk energy density f, the proof will be given in detail in the case when m= 1, f(∇u) = a|∇u|² andϕ(z) =b|z|/2 (so thatϕ(ν) +ϕ(−ν) =bif|ν|= 1) not to overburden notation, while the extension of the proof to the general case is given at the end of each section.

3. The lower inequality

For everyu∈L¹(Ω) andE measurable subset of Ω,we define

F(u, E) =

⎧⎪

⎪⎨

⎪⎪

⎩ a

Ω\E|∇u|²dx+b

2Hⁿ⁻¹(Ω∩∂E) if u∈H¹(Ω) and∂E Lipschitz

+∞ otherwise,

(1)

wherea, bare positive parameters. For this choice ofF Theorem 1 reads as follows.

Theorem 2. The lower-semicontinuous envelope of the functionalF with respect to theL¹(Ω)×L¹(Ω)topology, is the functionalF:L¹(Ω)× M(Ω)→[0,+∞]defined as:

F(u, E) =

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩ a

Ω\E|∇u|²dx+b

2Hⁿ⁻¹(∂^∗E∩Ω) +bHⁿ⁻¹(S(u)∩Ω∩E₀) if uχ_E0 ∈GSBV(Ω)

+∞ otherwise,

where∂^∗E is the reduced boundary of E andE₀ is the set of the points whereE has density0.

In order to prove a lower bound for the relaxation ofF we will use the ‘slicing’ method (see [12], Chap. 15) that allows to reduce to the study of energies defined on one-dimensional sections. To this end we will need to define as customary the ‘localized’ versions of our energies as follows. For everyu∈L¹(Ω), E measurable subset of Ω,and for everyA open subset of Ω,we define

F(u, E;A) =

⎧⎪

⎪⎨

⎪⎪

⎩ a

A\E|∇u|²dx+b

2Hⁿ⁻¹(∂E∩A) ifu∈H¹(Ω) and∂E Lipschitz

+∞ otherwise.

Similarly, we defineF(u, E;A).

Setting, for everyu∈L¹(Ω), E∈ M(Ω) andAopen subset of Ω:

F(u, E;A) = inf lim inf

j→∞ F(u_j, E_j;A) : u_j→u, E_j→E inL¹(Ω) , we show the following inequality:

F(u, E;A)≥F(u, E;A). (2)

This corresponds to proving the following

Proposition 3. Let u ∈L¹(Ω), E ∈ M(Ω) and A be an open subset of Ω. For every sequence{(u_j, E_j)} in L¹(Ω)× M(Ω), such thatu_j→uandE_j →E inL¹(Ω):

lim inf

j→+∞F(u_j, E_j;A)≥F(u, E;A).

Proof. We start by stating the result concerning the one-dimensional functionals.

– The lower inequality in the1-dimensional case

LetI⊂Rbe an interval,u∈L¹(I) and letE be a measurable subset ofI.Let{u_j}be a sequence inL¹(I), and let{E_j}be a sequence of measurable subsets ofI, such that

u_j→u in L¹(I) and E_j →E as j→+∞.

We can assume sup_jF(u_j, E_j;I)≤c; then,u_j∈H¹(I), the number of the connected components ofE_j is uni- formly bounded, and we can find a finite setS={s₁, . . . , s_L} ⊂E₀,a finite set of intervals{[a1, b₁], . . . ,[a_M, b_M]}

and a subsequence (not relabelled) such that

∀η >0 ∃j∈N ∀j≥j: E_j ⊂

L i=1

[s_i−η, s_i+η]∪

M i=1

[a_i−η, b_i+η].

In particular, we can assumeE=_M

i=1[a_i, b_i].Then, setting Λ_η =_L

i=1[s_i−η, s_i+η] andE_η=_M

i=1[a_i−η, b_i+η]:

I\(Λη∪Eη)

|u_j|²dx≤c;

by the arbitrariness ofη > 0, it follows that u∈H¹((I\E)\S),henceu∈SBV(I\E) and S(u)∩E₀ ⊂S.

This allows to conclude

lim inf

j→+∞F(u_j, E_j;I)≥F(u, E;I). (3)

It is easy to check that for everyJ open subset ofI:

lim inf

j→+∞F(u_j, E_j;J)≥F(u, E;J). (4)

– The lower inequality in then-dimensional case

We recall some definitions and properties related to the slicing procedure (for a general overview of this method we refere.g., to [13], Sect. 3.4 or [11], Sect. 4.1). For everyξ∈Sⁿ⁻¹let Π_ξ be the (n−1)-dimensional linear subspace ofRⁿ orthogonal to ξ. IfB⊆Rⁿ thenB^ξ be the orthogonal projection ofB on Π_ξ. For every y∈B^ξ setB^ξy={t∈R: y+tξ∈B}. Iff:B→Rletf^ξy:B^ξ,y →Rbe defined byf^ξ,y(t) =f(y+tξ).

The following results hold [5]:

(i) Letu∈GSBV(Ω) and letD^kustand for any ofD^a, D^jorD^c(the absolutely continuous, jump or Cantor part of the derivative). Then, for everyξ∈Sⁿ⁻¹and forHⁿ⁻¹-a.e. y∈Ω^ξ we haveu^ξy ∈GSBV(Ω^ξy);

moreover, denoting byD^ku, ξthe component ofD^kualongξ,the following representation holds:

BD^ku, ξ=

B^ξ

D^ku^ξy(B^ξy)dHⁿ⁻¹(y).

(ii) Letu∈L¹(Ω); assume that u^ξy∈SBV(Ω^ξy) for everyξin a basis ofRⁿ and fora.e. y∈Ω^ξ,and that

Ω^ξ

|Du^ξy|(Ω^ξy)dHⁿ⁻¹(y)<+∞.

Then u∈SBV(Ω).Before proceeding in the proof we recall a result concerning the supremum of a family of measures (see [12], Lem. 15.2).

Proposition 4. Let Ω be an open subset of Rⁿ andµ a finite, positive set function defined on the family of open subsets ofΩ.Letλbe a positive Borel measure onΩ,and{g_i}i∈Na family of positive Borel functions onΩ.

Assume that µ(A)≥

Ag_idλfor every A and i, and thatµ(A∪B)≥µ(A) +µ(B)whenever A, B ⊂⊂Ωand A∩B=∅ (superadditivity). Then µ(A)≥

A

sup_ig_i) dλfor everyA.

Moreover, we will use the following property.

Lemma 5. LetΓ⊂E₀ be a(n−1)-rectifiable subset,ξ∈Sⁿ⁻¹ such thatξ is not orthogonal to the normalν_Γ toΓ at any point ofΓ; then, for almost every y∈Π_ξ,for every t∈Γ^ξy, E^ξy has density0 in t.

Proof of Lemma 5. Since Γ is contained in a countable union of C¹ hypersurfaces, up to localization on one of those surfaces and a deformation argument, we can assume Γ⊂Π_ξ.

We set:

Γ_k =

y∈Γ : Θ(0, E^ξy)≥1 k

, Γ⁺=

k∈N

Γ_k,

where Θ(0, E^ξy) denotes the (one-dimensional) density of the setE^ξy in 0. In the following Q^ξ_ρ(x) denotes a cube with centrex, side ρ, and two parallel faces orthogonal toξ.

Let us assume by contradiction, thatHⁿ⁻¹(Γ⁺)>0. Then, there existsk such that Hⁿ⁻¹(Γ^k)>0. Since Γ_k⊂E₀,Lebesgue’s dominated convergence theorem gives

→0lim

Γk

|E∩Q^ξ(x)|

n dHⁿ⁻¹(x) = 0.

Then

0 = lim inf

→0

1

n−1

Γk

Γk∩Q^ξ(x)

|E^ξy∩(− /2, /2)|

dHⁿ⁻¹(y) dHⁿ⁻¹(x)

= lim inf

→0

1

n−1

Γk

Γk∩Q^ξ(y)

|E^ξy∩(− /2, /2)|dHⁿ⁻¹(x) dHⁿ⁻¹(y)

= lim inf

→0

Γk

1

n−1|Γ_k∩Q^ξ(y)||E^ξy∩(− /2, /2)|

dHⁿ⁻¹(y)

≥

Γk

lim inf

→0

|E^ξy∩(− /2, /2)|

dHⁿ⁻¹(y)≥Hⁿ⁻¹(Γ_k)

k ·

This gives the contradiction.

Now we apply the slicing method to complete the proof of Proposition 3, in a fashion similar to that in [9].

Letφandφdenote the one dimensional versions of the functionalsF andF, respectively. For everyξ∈Sⁿ⁻¹ we defineF^ξ:L¹(Ω)× M(Ω)× A(Ω)→[0,+∞] as

F^ξ(v, B;A) =

Πξ

φ(v^ξ,y, B^ξ,y;A^ξ,y) dHⁿ⁻¹(y).

Note thatF≥F^ξ for everyξ.

An application of the Fatou Lemma and the one dimensional inequality (4) give F(u, E;A)≥F^ξ(u, E;A),

where, forv∈L¹(Ω) andB∈ P(Ω):

F^ξ(v, B;A) =

Πξ

φ(v^ξy, B^ξy;A^ξy) dHⁿ⁻¹(y).

Thus, ifF(u, E;A) is finite, it follows thatE^ξy is a finite union of intervals which we can suppose closed, and that u^ξy∈SBV(A^ξy\E^ξy) for a.a. y in Π_ξ; moreover

Πξ A^ξy\E^ξy|(u^ξy)|²dt+H⁰((S(u^ξy)∩(E^ξy)₀∩A^ξy))

dHⁿ⁻¹(y)<+∞.

Then, if in additionu∈L^∞(Ω), we get

Πξ

|D(u^ξy)|(A^ξ,y\E^ξy)dHⁿ⁻¹(y)<+∞.

Recalling (ii), we deduce that, assumingu∈L^∞(Ω),ifF(u, E;A) is finite, thenuχ_E0 ∈SBV(A).A truncation argument allows to conclude thatF(u, E;A) is finite only ifuχ_E0 ∈GSBV(A); in order to conclude applying (i), we need to prove thatS(u^ξy)∩(E₀)^ξy ⊂(S(u^ξy)∩E^ξy)₀; this follows from Lemma 5, with Γ =S(u)∩E₀∩ {ξ, ν_u = 0}Now, recalling (i), we get:

F^ξ(u, E;A) =a

A|∇u, ξ|²dx+b

A∩S(u)∩E0

|ν_u, ξ|dHⁿ⁻¹+b 2

A∩∂^∗E|ν_E, ξ|dHⁿ⁻¹.

Since, ifuχ_E0∈GSBV(A),the set functionF(u, E;·) is superadditive on disjoint open sets, an application of Proposition 4 withλ=Lⁿ+Hⁿ⁻¹ (S(u)∩E₀) +Hⁿ⁻¹ ∂^∗E,and

g_i(x) =

⎧⎨

⎩

a|∇u, ξ_i|² if x∈Ω\

(S(u)∩E₀)∪∂^∗E b|ν_u, ξ_i| if x∈S(u)∩E₀

b

2|ν_E, ξ_i| if x∈∂^∗E,

where{ξ_i} is a dense sequence inSⁿ⁻¹ such thatξ_i, ν_u = 0Hⁿ⁻¹-a.e. onS(u)∩E₀, gives F(u, E;A)≥F(u, E;A)

as desired.

Remark 6. The same proof allows to treat the case whenm≥1

F(u, E;A) =

⎧⎪

⎪⎨

⎪⎪

⎩ c

A\E∇u^pdx+

∂E∩A

ϕ(ν_E)dHⁿ⁻¹ ifu∈W^1,p(Ω;R^m), ∂ELipschitz

+∞ otherwise

where ∇u= sup{∇u·ξ:|ξ|= 1} andc is a positive constant, obtaining the lower inequality withF as in the thesis of Theorem 1. The necessary modifications to the slicing procedure are standard and can be found in [11], Section 4.1.2.

Remark 7. For general quasiconvexf as in Theorem 1 we may consider the lower semicontinuous functional [5]

onGSBV(Ω;R^m) defined by

G(v;A) =

A

f(∇v) dx+cHⁿ⁻¹(S(v)∩A).

Ifu_j→uandE_j→E we have, settingv_j =u_j(1−χ_Ej) andv=u(1−χ_E) lim inf

j F(u_j, E_j;A) ≥ lim inf

j

G(v_j, A)− |A∩E_j|f(0)

≥ G(v;A)− |A∩E|f(0)

=

A

f((1−χ_E)∇u) dx+cHⁿ⁻¹(S(v)∩A)− |A∩E|f(0)

≥

A\E

f(∇u) dx.

Note that this identification ofF withGgives a lower bound that is optimal for the bulk term but not for the surface energy.

Remark 8(Proof of the lower bound in the general case). Let nowF be defined as in Theorem 1 and letuand E be such that uχ_E0 ∈GSBV(Ω;R^m) and F(u, E;A)<+∞. By the growth conditions onf and Remark 6 we then have

F(u, E;A)≥

A∩∂^∗E

ϕ(ν_E) dHⁿ⁻¹+

A∩S(u)∩E0

(ϕ(ν_u) +ϕ(−ν_u)) dHⁿ⁻¹; by Remark 7 on the other hand we obtain

F(u, E;A)≥

A\E

f(∇u) dx.

We can define apply Proposition 4 withµ(A) =F(u, E;A), the measureλdefined byλ(A) =|A|+Hⁿ⁻¹(A∩ (S(u)∩E₀)) +Hⁿ⁻¹(A∩∂^∗E), g₁(x) =f(∇u)χ_E0\S(u) andg₂(x) =ϕ(ν_E)χ_∂∗E+ (ϕ(ν_u) +ϕ(−ν_u))χ_S(u)∩E0, to obtain the lower inequality in the general case.

4. The upper inequality

In order to give an upper estimate, we introduce, for everyu∈L¹(Ω), E ∈ M(Ω) andAopen subset of Ω:

F(u, E;A) = inf lim sup

j→+∞ F(u_j, E_j;A) : u_j→u, E_j→E inL¹(Ω), .

The proof of Theorem 2 is complete if we show that the following inequality holds for every u ∈ L¹(Ω), E∈ M(Ω) andAopen subset of Ω:

F(u, E;A)≤F(u, E;A). (5)

It is clearly sufficient to prove the following:

Proposition 9. Let u ∈ L¹(Ω), E ∈ M(Ω) such that uχ_E0 ∈ GSBV(Ω). Then, there exists a sequence {(uj, E_j)} ∈H¹(Ω)× M(Ω), with∂E_j of classC^∞, such that u_j→u, E_j →E inL¹(Ω), and

lim sup

j→+∞ F(u_j, E_j)≤F(u, E).

In order to construct the recovery sequence, let us recall the definition of strong convergence in SBV^p, introduced in [14], and an approximation lemma forSBV^p with piecewiseC¹ functions.

Definition 10. ([14]) Let{u_j}be a sequence of functions inSBV^p. We say thatu_j converges strongly touin SBV^p if

– u_j→uinL¹(Ω),

– ∇u_j → ∇ustrongly inL^p(Ω;Rⁿ), – Hⁿ⁻¹(S(u_j)∆S(u))→0,

–

S(uj)∪S(u)

(|u⁺_j −u⁺|+|u⁻_j −u⁻|)dHⁿ⁻¹→0

(we choose the orientationν_uj =ν_uHⁿ⁻¹-a.e.onS(u_j)∩S(u); recall that ifv∈BV(Ω) then we setv⁺ =v⁻=v on Ω\S(v)).

Lemma 11. [14] If u ∈ SBV^p(Ω)∩L^∞(Ω) then there exists a sequence {u_j} in SBV^p(Ω)∩L^∞(Ω) with u_j∞ ≤ u_∞, strongly converging to u in SBV^p, such that for each j ∈N there exists a closed rectifiable set R_j such that u_j ∈C¹(Ω\R_j). Moreover, R_j can be chosen so that its Minkowski content coincides with Hⁿ⁻¹(R_j); i.e.,

Hⁿ⁻¹(R_j) = lim

ρ→0⁺

1

2ρ|{x: dist(x, R_j)< ρ}|.

The proof of this Lemma in [14] consists in finding first a compact setK⊂S(u) such thatHⁿ⁻¹(S(u)\K)1 (which will be the main part ofR_j) and then then approximatinguon Ω\Kby minimizing a Mumford-Shah type functional.

In the following Mⁿ⁻¹(B) stands for the Minkowski content of a set B. As an intermediate step in the construction of the recovery sequence{(uj, E_j)},we apply the approximation result of Lemma 11 to prove the following:

Lemma 12. Let E be a set of finite perimeter withE = Ω\E₀ andu∈L^∞(Ω), such thatuχ_E0 ∈SBV(Ω).

Then, there exist a sequence of closed rectifiable sets {S_j}, a sequence of measurable subsets {F_j}, with ∂F_j closed rectifiable and Lipschitz in Ω\S_j, and a sequence of functions {w_j} in SBV(Ω)∩C¹(Ω\(S_j∪∂F_j)) such that

– w_j→uandF_j→E inL¹(Ω);

– Hⁿ⁻¹(S_j\(S(w_j)∩(F_j)₀)) =o(1)_j→+∞;

– Hⁿ⁻¹(∂F_j) =Mⁿ⁻¹(∂F_j), Hⁿ⁻¹(S_j) =Mⁿ⁻¹(S_j);

– lim sup

j→+∞ F(w_j, F_j)≤F(u, E).

Proof. We can supposeu_∞= 1; we set u:=

u in Ω\E 2 in E.

From Lemma 11, for everyj there exist a closed rectifiable setR_j and a functionu_j∈C¹(Ω\R_j),such thatu_j strongly converges tou,andHⁿ⁻¹(R_j\S(u_j))→0.

Setting v_j = (u_j∨1)∧2 andv = (u∨1)∧2, it follows that the sequence{v_j} strongly converges to v; in particular,

|Dv_j|(Ω) =

Ω

|∇v_j|dx+

S(vj)

|v_j⁺−v_j⁻|dHⁿ⁻¹→

Ω

|∇v|dx+

S(v)|v⁺−v⁻|dHⁿ⁻¹, and

|Dv|(Ω) =Hⁿ⁻¹(S(v)) =Hⁿ⁻¹(∂^∗E).

The coarea formula gives, forAopen subset of Ω:

Hⁿ⁻¹(∂^∗E∩A) = lim

j→+∞

₂

1

Hⁿ⁻¹(∂^∗E_j^t∩A) dt

whereE_j^t={x∈A : u_j ≥t}.Fixingδ >0,there existst_j∈(1,2−δ) such that, settingF_j=E_j^tj, Hⁿ⁻¹(∂F_j∩A)≤ 1

1−δHⁿ⁻¹(∂E∩A) (6)

and∂F_j is Lipschitz in Ω\R_j.

Now, we define the sequence{w_j}as:

w_j :=u_jχ_(Fj)0;

it follows thatw_j ∈C¹(Ω\(R_j∪∂F_j)),and setting S_j=R_j∩(F_j)₀we get Hⁿ⁻¹(S_j\(S(w_j)∩(E_j)₀)) =o(1)_j→+∞. Moreover:

Ω

|∇u|²dx≤lim inf

j→+∞

Ω

|∇w_j|²dx≤lim inf

j→+∞

Ω

|∇u_j|²dx=

Ω

|∇u|²dx=

Ω

|∇u|²dx, and this implies the convergence

∇w_j → ∇u in L²(Ω). (7)

The strong convergenceu_j→uentails:

Hⁿ⁻¹(∂E) +Hⁿ⁻¹(S(u)\E) =Hⁿ⁻¹(S(u)) = lim

j→+∞Hⁿ⁻¹(S(u_j))

= lim

j→+∞

Hⁿ⁻¹(S(u_j)\F_j) +Hⁿ⁻¹(S(u_j)∩F_j)

= lim

j→+∞

Hⁿ⁻¹(S(w_j)\F_j) +Hⁿ⁻¹(S(u_j)∩F_j)

. (8)

Again from the strong convergenceu_j→uwe obtain, in particular:

j→+∞lim

∂E\(S(uj)∩Fh)

(|1−(u_j)⁻|+|2−(u_j)⁺|) dHⁿ⁻¹= 0;

since|1−(u_j)⁻|+|2−(u_j)⁺| ≥δin ∂E\(S(u_j)∩F_j),it follows that

j→+∞lim Hⁿ⁻¹(∂E\(S(u_j)∩F_j)) = 0.

Then, we obtain

Hⁿ⁻¹(∂E) = lim

j→+∞Hⁿ⁻¹(∂E∩(S(u_j)∩F_j))

≤ lim inf

j→+∞Hⁿ⁻¹(S(u_j)∩F_j).

This inequality, taking into account (8), allows to deduce that:

lim sup

j→+∞ Hⁿ⁻¹(S(w_j)\F_j)≤ Hⁿ⁻¹(S(u)\E) =Hⁿ⁻¹(S(u)\E), (9)

concluding the proof of the lemma.

Proof of Proposition 9. To prove Proposition 9, we consider a functionuand a setEsuch thatF(u, E) is finite.

A truncation argument allows us to supposeu∈L^∞(Ω).Now, we consider the sequences{w_j},{F_j} and{S_j} given by Lemma 12. From the coarea formula, recalling thatHⁿ⁻¹(S_j) =Mⁿ⁻¹(S_j) and|∇dist(·, S_j)|= 1a.e., for fixedj it follows:

k _1/k

0

Hⁿ⁻¹(∂^∗{x∈Ω : dist(x, S_j)< r}) dr=

{dist(·,Sj)<1/k}|∇dist(x, Sj)|dx

=k

x∈Ω : dist(x, S_j)< 1

k= 2Hⁿ⁻¹(S_j) +o(1)_k→+∞ (10) so that there existsr^k_j ∈(0,1/k) with, letting Σ^k_j ={x∈Ω : dist(x, S_j)< r_j^k},

Hⁿ⁻¹(∂Σ^k_j)≤2Hⁿ⁻¹(S_j) +o(1)_k→+∞. Upon choosing a suitable sequencek_j and defining Σ_j= Σ^k_j^j, we then have

Hⁿ⁻¹(∂Σ_j)≤2Hⁿ⁻¹(S_j) +o(1)_j→+∞. Now, setting

E_j =F_j∪Σ_j, sinceHⁿ⁻¹(S_j) =Hⁿ⁻¹(S(w_j)∩E₀) +o(1)_j→+∞,we get:

Hⁿ⁻¹(∂E_j)≤ Hⁿ⁻¹(∂F_j) + 2Hⁿ⁻¹(S(w_j)) +o(1)_j→+∞. (11) For every E_j, it is easy to show that there exists a set E_j of classC^∞ such thatE_j ⊂E_j andHⁿ⁻¹(∂E_j) = Hⁿ⁻¹(∂E_j) +o(1),so we can assumeE_j of classC^∞. Then we can find, for everyj,a function ˜u_j∈H¹(Ω) such that the restriction ofu_jto the set (E_j)₀coincides with the restriction ofw_j.We then setu_j =φ_ju˜_j+(1−φ_j)v_j, wherev_j are smooth functions converging touinL¹(Ω), andφ_j are smooth functions withφ_j= 1 on (E_j)₀and φ_j(x) = 0 if dist(x,(E_j)₀)>1/(2r_j). Clearly, the sequence{u_j} converges touin L¹(Ω); the inequality (11) implies:

lim sup

j→+∞ F(u_j, E_j)≤lim sup

j→+∞ F(w_j, E_j),

and this completes the proof of the upper inequality and of the Proposition 9.

Remark 13. An alternative proof of the upper inequality would consist in first approximating the set E as follows: sinceχ_E∈BV(Ω), by standard results [21] there exists a sequence{v_j} ⊂C^∞(Ω), converging toχ_Ein L¹(Ω) and such that

Ω|∇v_j| → |Dχ_E|=Hⁿ⁻¹(∂^∗E). It is easy to show that for alls∈(0,1), χ_{vj≥s}→χ_E in L¹(Ω).

Moreover, for a.a.s∈(0,1), ∂{v_j≥s} ∩Ω isC^∞ (by Sard’s lemma), and using the coarea formula we find Hⁿ⁻¹(∂{v_j≥s} ∩Ω)→ Hⁿ⁻¹(∂^∗E).

From the construction in [21] (made by locally convolvingχ_Ewith suitable mollifiers) we also can assume that

v_j(x)→

⎧⎨

⎩

0 in E₀ 1 in E₁

1

2 in E_1/2 Hⁿ⁻¹-a.a. x,

whereE_1/2 is the set of points where Ehas density ¹₂.Hence, for s <¹₂,it follows that:

χ_{vj≥s}(x)→

0 in E₀

1 in Ω\E₀, Hⁿ⁻¹-a.a. x, recalling thatHⁿ⁻¹(Ω\(E₀∪E₁∪E_1/2)) = 0; in particular, fors < ¹₂:

∂^∗E

χ_{vj≥s}(x) dHⁿ⁻¹→ Hⁿ⁻¹(∂^∗E).

Then, forε >0,we can chooses < ¹₂ such that, settingF_j={v_j≥s}:

Hⁿ⁻¹(∂^∗E\F_j)< ε for j large enough. (12) Now, letu∈SBV(Ω). We letu_j=u|_Ω\Fj, which is viewed as a function defined in the open set Ω\F_j. Clearly u∈SBV(Ω\F_j), and one has

S(u_j) ⊂ (∂^∗E\F_j)∪(S(u)∩E₀), so that

Hⁿ⁻¹(S(u_j)) ≤ Hⁿ⁻¹(S(u)∩E₀) + ε. (13)

By standard approximation results, for everyj there exists u_j ∈ SBV(Ω\F_j) with u_j−u_jL¹ < 1/j, and such that: Hⁿ⁻¹(S(u_j)\S(u_j)) = 0 (so thatHⁿ⁻¹(S(u_j)) =Mⁿ⁻¹(S(u_j))), u_j ∈C¹((Ω\F_j)\S(u_j)), and

Hⁿ⁻¹(S(u_j)) ≤ Hⁿ⁻¹(S(u_j)) + ε.

Recalling (12), this inequality and (13), we find:

Hⁿ⁻¹(S(u_j)) ≤ Hⁿ⁻¹(S(u)∩E₀) + 2ε.

We can now select an appropriate neighborhood Σ_j of S_j =S(u_j) as in the previous proof, and letting E_j = F_j∪Σ_j, we conclude as before.

Remark 14 (Proof of the upper bound in the general case). The proof is not much modified in the general case since the main difficulty is the construction of the approximating sets, which is independent ofmand of the particular energy.

Some care must be taken while following the reasoning in (10). In the case whenϕ is even, then standard results on the anisotropic Minkowski contents (where the distance function is replaced byϕ^◦ :x→sup_ϕ(ξ)≤1 ξ·x, the polar of ϕ, see [6]) allow to easily adapt the previous proof to the anisotropic case. However, the nonsymmetric case is not covered by these results (although it is very likely that they extend to this situation).

We first note that if S_j is composed of a finite number of compact subsets of C¹ hypersurfaces, by the conditionMⁿ⁻¹(S_j) =Hⁿ⁻¹(S_j) we have

k

x∈Ω : dist(x, S_j)< 1

k=k

y+tν(y) :y∈S_j,|t|< 1

k+o(1)_k→+∞ (14)

(but in the second representation, the same point might correspond to two or more values ofyandt). We then have

k _1/k

0

∂^∗{x∈Ω:dist(x,Sj)<r}ϕ(∇dist(x, S_j)) dHⁿ⁻¹

=

{dist(·,Sj)<1/k}ϕ(∇dist(x, Sj))|∇dist(x, Sj)|dx

≤ k

Sj

_1/k

−1/kϕ(∇dist(y+tν(y), S_j)) dtdHⁿ⁻¹(y) +o(1)_k→+∞

=

Sj

(ϕ(ν) +ϕ(−ν)) dHⁿ⁻¹+o(1)_k→+∞

sinceHⁿ⁻¹-a.e.onS_j,

t→0lim^±∇dist(y+tν(y), S_j) =±ν(y).

If we define Σ^k_j as before, we find that lim sup

k→∞

∂^∗Σ^k_j

ϕ(∇dist(x, Sj)) dHⁿ⁻¹= lim sup

k→∞

∂^∗Σ^k_j

ϕ(ν_Σk

j) dHⁿ⁻¹ ≤

Sj

(ϕ(ν) +ϕ(−ν)) dHⁿ⁻¹

and are able to carry on the proof of Proposition 9. In the general case we can split S_j in a part composed of a finite union of compact subsets ofC¹ hypersurfaces and a remainder whose Minkowski content is arbitrarily small, and proceed likewise.

The only technical point where some extra care must be used is the truncation argument at the beginning of the proof of Proposition 9. That argument is straightforward in the scalar case while in the vector case some more elaborate but by now standard truncation lemmas must be used (for example [17], Lem. 3.5).

It remains to prove that if 0 < |E| ≤Ω then we can find a recovery sequence with |E_j| = |E|. The case

|E| = Ω is trivial. In the case 0 <|E|<Ω we can simply modify E_j by inserting or removing suitable balls with suitable volume close to||E| − |E_j||(and possibly smoothing the resulting sets if needed). The location of the centres of such balls must be chosen as a point of density 0 or 1 forE, respectively. Details can be found, e.g., in [2], Theorem 3.3.

Remark 15(An extension). As suggested by an anonymous referee, the results of Theorem 1 can be extended to a more generalF of the form

F(u, E) =

Ω\E

f(∇u) dx+

∂E

ϕ(u⁻, ν_E)dHⁿ⁻¹,

where u⁻ denotes the trace of the restriction of uto Ω\E on ∂E. If ϕ: Ω×Rⁿ →[0,+∞) is a continuous function such that ϕ(u,·) is convex and positively homogeneous of degree one and ϕ(u, z)≥c|z|with c >0, then the thesis of Theorem 1 still holds with

F(u, E) =

Ω\E

f(∇u) dx+

∂^∗E

ϕ(u⁻, ν_E)dHⁿ⁻¹+

Ω∩S(u)∩E0

(ϕ(u⁻, ν_u) +ϕ(u⁺,−ν_u))dHⁿ⁻¹,

with the usual notation for the traces ofuon both sides ofS(u).

The proof of the lower bound can be achieved in the same way, noting that the lower bound for the bulk part (Rem. 7) remains unchanged, while we first easily deal with one-dimensional integrands of the formϕ(u⁻), subsequently obtaining the lower bound by slicing. In view of Remark 7 is is worth noting that, even in the