Approximation result for a Mumford-Shah functional adapted to a segmentation problem

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Approximation result for a Mumford-Shah functional adapted to a segmentation problem

David Vicente

To cite this version:

David Vicente. Approximation result for a Mumford-Shah functional adapted to a segmentation problem. 2014. �hal-01087205v2�

Approximation result for a Mumford-Shah functional adapted to a segmentation problem

David Vicente November 28, 2014

Abstract

We consider a segmentation problem of image close to a binary one. After introducing a Mumford Shah energy like, we perform an approximation of this model. We prove a Γ-convergence result and show why this approximation is suitable for numerics.

1 Introduction

This paper is motivated by a 3D-image segmentation process with thin tubu- lar structures. We have introduced a model which is derived from Mumford and Shah’s one in [7]. Mathematical analysis of such model is difficult. So, in order to make numerical experiments, we perform an approximation in the sense of Γ-convergence. More precisely, an image is modeled as an ap- plicationg defined on Ω with values on [0,1] where Ω is an open, bounded and regular subset of Rⁿ. We assume that g is close to a binary function.

More precisely, its histogram has two distincts modes that we assume to be 0 and 1. We prove that the model

(P): Min{E(p): p∈SBV(Ω), p(x)∈ {0; 1} a.e. x∈Ω},

whereE(p) = ¹₂^R_Ω(p−g)²dx+βHⁿ⁻¹(S_p). (1.1) whereSp is the jump set of the binary functionp, may be approximated by

(P_ε): MinE_ε(p):p∈W^1,2(Ω),0≤p(x)≤1 a.e. x∈Ω

whereE_ε(p) = ¹₂^R_Ω(p−g)²dx+β^R_Ω9ε|∇p|²+^p2(1−p)_ε ²dx (1.2) in the sense of the Γ-convergence. In the application which motivated this paper, we setn= 3 but we prove this result for any n≥1.

In a companion paper [4] we perform a tuning for the parameters of this approximated model in order to detect thin structures of tubes in 3D images and we give there a complete bibliography of the subject. In the present paper we introduce the theoretical framework (section2) then we prove the Γ-convergence of the model (section3) and give additional properties related to numerical aspects (section4).

2 Functional framework and Γ-convergence

2.1 Functional spaces

We work in this paper with spaces of functions with bounded variation that we recall thereafter, for more details see [5], [1].

Definition 2.1. Let Ω be an open subset Rⁿ, and u ∈ L¹(Ω). We define BV(Ω), the space of functions with bounded variations:

BV(Ω) ={u∈L¹(Ω): T V(u)<+∞}, where

T V(u) = sup^Z

Ω

u(x)div(ξ)(x)dx:ξ∈C_c¹(Ω),kξk_∞≤1.

The space BV(Ω), with the norm kuk_BV(Ω) = kuk_L1 +T V(u), is a Banach space. The derivative, in a distributional sense, of an element u ∈ BV(Ω) is a Radon measure, denotedDu, and T V(u) =^R_Ω|Du|is the total variation of u. We introduce the following notation for a measure µ and a setB,µ ¬

B is the measure such that:

µ ¬

B(A) =µ(A∩B).

We recall useful properties on the structure of the gradient ofBV func- tions in next theorem.

Theorem 2.1. Let u∈BV(Ω) then we have the following decomposition:

Du=∇u· L_n+ (u⁺−u⁻)ν_uHⁿ⁻¹ ¬

S_u+C_u, where

• ∇u= ^Du_Ln,

• Su is a countable union of hypersurfaces,

• ν_u:S_u→Sⁿ⁻¹ is a measurable normal unitary vector of S_u,

• u⁻, u⁺ are the lower and upper approximated limits ofu,

• C_u ⊥ Hⁿ⁻¹.

Definition 2.2. For any u∈BV(Ω), we set:

• ∇u· L_n as the regular part of Du,

• Su as the jump set of u,

• (u⁺−u⁻)ν_uHⁿ⁻¹ ¬

S_u as the jump part of Du,

• C_u as the Cantor part of Du.

Definition 2.3. The set of BV-functions whose derivative have a Cantor part equal to zero is called the special set of functions with bounded variation, denoted SBV(Ω).

Theorem 2.2. Let u = 1_A an indicator function such that u ∈ SBV(Ω) then we have the following decomposition:

Du=ν_uHⁿ⁻¹ ¬ S_u. The space SBV has a useful slicing property.

Definition 2.4. Let ν ∈Sⁿ⁻¹ (unit sphere of Rⁿ) and Ω be an open subset of Rⁿ. Let Πν be the orthogonal plane toν:

Πν ={x∈Rⁿ:hν, xi= 0},

where h·,·i stands for the scalar Rⁿ-product. For any x∈Πν we define Ωx={t∈R:x+tν ∈Ω}

and Ων ={x∈Πν : Ωx6=∅}, see figure2.1.

Figure 2.1: Slicing the open Ω

As in [2], we introduce a projection on the open set Ωx (see figure 2.1).

For any functionu defined on Ω⊂Rⁿ and any x∈Ων, we set u_x: Ωx → R

t → u(x+tν)

Theorem 2.3. Let u∈L^∞(Ω)be a function such that, for all ν ∈Sⁿ⁻¹, i) ux∈SBV(Ωx) for Hⁿ⁻¹ a.e. x∈Ων,

ii) ^R_Ων^nR_Ωx|∇u_x|dt+H⁰(S_ux)^odHⁿ⁻¹(x)<+∞where S_v is the jump set of the function v (see 2.2);

then,u∈SBV(Ω)and Hⁿ⁻¹(S_u)<+∞.

Conversely, let u ∈SBV(Ω)∩L^∞ be such that Hⁿ⁻¹(S_u) <+∞. Then i) and ii) are satisfied. Moreover, we have

iii) h∇u(x+tν), νi=∇u_x(t), for a.e. t∈Ωx and Hⁿ⁻¹-a.e. x∈Ων, iv) there exists a measurable function ν_u : S_u → Sⁿ⁻¹ depending on S_u,

such that Z

Su

hν_u, νidHⁿ⁻¹(x) =^Z

Ων

H⁰(Sux)dHⁿ⁻¹(x). 2.2 Γ-convergence theory

Usually, a standard compactness argument is sufficient to prove the existence of minimizing solutions. In our case, we use a specific notion of convergence which will be specified here, since compactness is not ensured for usual topologies. As before, for more details see [3].

Definition 2.5. Let (X, d) a metric space,(Fn)n∈N a sequence of functions defined onX with values inR∪{+∞}andF:X→R∪{+∞}. The sequence (F_n)n∈N Γ-converges to F on x∈X if the following assertions are true:

i) for all (x_n)n∈N which converges to x:

F(x)≤lim inf_n→∞ Fn(xn), ii) there exists (yn)n∈N which converges to x such that:

F(x)≥lim sup

n→∞ F_n(y_n),

This concept is the convergence of the approximate minimum. More precisely, if for anyn,pnis a minimum ofFnand (pn)nconverges top, then according to this concept,p is a minimum ofF.

We shall use the following result.

Proposition 2.1. Let X a topological space, (f_n)n∈N a sequence of func- tions defined on X with values in R∪ {+∞} which Γ-converges to f and h a continuous function defined on X with values in R∪ {+∞}. Then the sequence(f_n+h)n∈N Γ-converges to f+h.

3 A Γ-convergence result

We now prove that the family of functionals (Eε)ε>0 defined by (1.2) Γ- converges toEdefined by (1.1). For more simplicity in the proof we assume that β = 1 in this section, its value doesn’t change the arguments of the proof. We denoteW_b^1,2(Ω) the following space

W_b^1,2(Ω) =ⁿp∈W^1,2(Ω): 0≤p(x)≤1 a.e. x∈Ω^o.

LetB(Ω; [0,1]) be the space of measurable functions defined on Ω which take their values in [0,1]. This space is endowed with the almost everywhere con- vergence topology. It’s a Fréchet space (see [2]).

LetF be the penalization term of the functional E:

F:B(Ω; [0,1]) → R∪ {+∞} (3.1) p 7→

( Hⁿ⁻¹(Sp) if p∈SBV(Ω),

+∞ otherwise (3.2)

and F_ε be its approximation:

F_ε:B(Ω; [0,1]) → R∪ {+∞}

p 7→

( R

Ω

9ε|∇p|²+^p2(1−p)_ε ²dx, ifp∈W_b^1,2(Ω),

+∞ otherwise.

LetH be the fitting data term:

H:B(Ω; [0,1]) → R∪ {+∞}

p → 1 2

Z

Ω(p−g)²dx.

Let (p_k)k∈N⊂B(Ω; [0,1]) be a a.e. converging sequence to somep. Accord- ing Lebesgue theorem and the fact thatkp_kk∞≤1, the sequence (H(pk))k∈N

converges to H(p) and H is continuous for the almost everywhere conver- gence topology. So, it is sufficient to prove the Γ-convergence of F_ε to F. Indeed, proposition 2.1with X =B(Ω; [0; 1]), implies the Γ-convergence of Eε=Fε+H toE=F+H.

We introduce the following notation:

Fε(p, U) =^Z

U 9ε|∇p|²+p²(1−p)² ε

!

dx, F(p, U) =Hⁿ⁻¹(Sp∩U) and

F−(p, U) =inflim inf

k→∞ Fε_k(p_k, U)|p_k^a.e.→ p,(p_k)k∈N⊂W_b^1,2(U). (3.3) The functionalF− is also called the lower-Γ limit of F.

We shall prove the following theorem.

Theorem 3.1. Let (ε_k)k∈N be a sequence which converges to 0⁺ and p ∈ SBV(Ω) be a function taking its values in{0,1}. Then

i)

F(p,Ω)≤F−(p,Ω) (3.4) holds;

ii) there exists a sequence (pk)k∈N ⊂ W_b^1,2(Ω) which converges a.e. to p such that:

F(p,Ω)≥lim sup

k→∞

F_εk(p_k,Ω). (3.5) According to this theorem, we deduce that F(p,Ω) = F−(p,Ω) for p ∈ SBV(Ω) taking its values in{0,1}. The proof is detailed thereafter.

3.1 The inequality for the lower Γ-limit (3.4)

We now prove the first inequality (3.4). We shall need the following prop- erties of this functional:

a) F−(p,·) is supperaddditive on open sets, that is

A∩B=∅ ⇒ F−(p, A∪B)≥ F−(p, A) + F−(p, B), b) F−(p,·) is non decreasing:

A⊂B ⇒ F−(p, A)≤F−(p, B).

By a diagonal extraction, there exists a sequence (p_k)k∈N⊂W_b^1,2(Ω) such

that: ₍

p_k^a.e.→ p,

Fεk(pk,Ω)→F−(p,Ω) (3.6) We assume that F−(p,Ω)<+∞ otherwise (3.4) is ensured. We have to prove:

F(p,Ω)≤lim inf

k→∞ F_εk(p_k,Ω)

As in [2], we perform the proof in two steps: the first step deals with dimension 1. The second generalizes it for dimensionn≥2.

3.1.1 The one-dimensional case

Assume Ω = I is an open interval. To avoid confusion, when n = 1, we denote the approximating functionals byF_ε¹ et their lower Γ-limit, defined in (3.3), byF₋¹.

Lemma 3.1. Let B_η(x)⊂Rbe an open ball andq ∈B(B_η(x))be a function taking its values in {0; 1}. Assuming

∀ρ∈]0, η[, q 6∈W^1,2(B_ρ(x)), then we have:

∀ρ∈]0, η[, F₋¹(q, B_ρ(x))≥1.

In particular, if I ⊂R is an open and bounded interval such that q ∈ B(I) takes its values in{0; 1} and F₋¹(q, I)<+∞ thenq ∈SBV(I). Moreover, there existsJ ⊂I with cardinal less than F₋¹(q, I) such that

i) p is constant on each connected component of I\J, ii) H⁰(Sq∩I)≤F₋¹(q, I).

Proof. We can assume thatF₋¹(q, B_ρ(x))<+∞for anyρ∈]0;η[. As before, there exists a sequence (q_k)k∈N⊂W_b^1,2(Bρ(x)) such that:

( q_k^a.e.→ q,

F_εk(q_k, B_ρ(x))→F₋¹(q, B_ρ(x)) (3.7) Sinceqtakes its values a.e. in{0,1}andq6∈W^1,2(B_ρ⁰(x)) for anyρ⁰ ∈]0, ρ[, there exists two sequences (y_k¹)k∈N and (y²_k)k∈N such that:

y¹_k→x, q_k(y_k¹)→1 andy_k²→x, q_k(y²_k)→0. However, we have:

F_ε¹_k(qk, Bρ(x))≥ Z x+ρ

x−ρ 9εk|∇q_k(y)|²+ q²_k(1−qk(y))² ε_k

! dy.

We introduce:

A:= 3√

εk|∇q_k(y)|, B := q_k(1−q_k(y))

√εk

. With the inequalityA²+B²≥2AB, we get:

F_ε¹_k(q_k, B_ρ(x))≥6^{Z x+ρ}

x−ρ

|∇q_k(y)|q_k(y)(1−q_k(y))dy.

As [y¹_k, y²_k]⊂B_ρ(x), we obtain:

F_ε¹_k(q_k, Bρ(x))≥6^{Z y}

2 k

y¹_k

|∇q_k(y)|q_k(y)(1−q_k(y))dy.

Sinceq_k ∈W^1,2(I), we may use the change of variablet=q_k(s). This yields:

F_ε¹

k(q_k, B_ρ(x)) ≥ 6^{Z qk(y}

2 k) qk(y_k¹)

t(1−t)dt, F_ε¹_k(qk, Bρ(x)) ≥ 6

q²_k(y¹_k)−q_k²(y_k²)

2 − q_k³(y_k¹)−q_k³(y_k²) 3

We know thatq_k(y_k¹)→1 andq_k(y²_k)→0, so that we deduce:

q²_k(y¹_k)−q_k²(y_k²)

2 − q_k³(y_k¹)−q_k³(y_k²)

3 → 1

6. It comes,

lim infF_ε¹_k(qk, Bρ(x))≥1.

and we can conclude

F₋¹(q, Bρ(x))≥1. (3.8) We set

J =ⁿx∈I:∀ρ >0, q6∈W^1,2(B_ρ(x))^o.

The functional F₋¹(q,·) is super-additive and non-decreasing (see (3.1)).

With inequality (3.8) we get

F₋¹(q, I)≥ H⁰(J).

According to the fact thatq takes its values a.e. in{0; 1}, thenqis constant on the connected components ofI \J and Sq∩I =J. Therefore, we may conclude

F₋¹(q, I)≥ H⁰(S_q∩I).

3.1.2 Generalization to dimensions n≥2

We now extend the proof to any higher dimensionnby a slicing argument.

Recall that we want to proveF−(p,Ω)≥F(p,Ω).

Let (p_k)k∈N⊂W_b^1,2(Ω) as in (3.6). LetAbe an arbitrary open subset of Ω.

Assume thatF−(p, A)<+∞ and F_εk(p_k, A) →F−(p, A). Let ν ∈Sⁿ⁻¹ be an unit vector, we have:

F_εk(p_k, A) = ^Z

A 9ε_k|∇p_k(x)|²+p²_k(x)(1−p_k(x))² εk

! dx

≥ Z

A 9ε_k|h∇p_k(x), νi|²+p²_k(x)(1−p_k(x))² ε_k

! dx

The Lebesgue measure ofRⁿprojected on Πν is the Hausdorff measure with dimensionn−1 (see for example [5]). Applying Fubini theorem, we get:

F_εk(p_k, A) ≥ Z

Aν

Z

Ax

9ε_k|h∇p_k(x+tν), νi|²+p_k(x+tν)²(1−p_k(x+tν))² ε_k

!

dt dHⁿ⁻¹(x)

≥ Z

Aν

F_ε¹_k((p_k)x, A_x) dHⁿ⁻¹(x). Fatou lemma yields:

F−(p, A) ≥ Z

Aν

lim infF_ε¹_k((p_k)x, Ax) dHⁿ⁻¹(x). As (p_k)x∈SBV(Ax) a.e. x∈Aν, with lemma3.1, we have:

F−(p, A) ≥ Z

Aν

H⁰(S(p)x∩Ax) dHⁿ⁻¹(x).

Using theorem2.3 gives:

Z

Aν

H⁰(S_(p)x ∩A_x) dHⁿ⁻¹(x) =^Z

Sp∩A

|hν_p, νi| dHⁿ⁻¹(x), so that

F−(p, A)≥ Z

Sp∩A

|hν_p, νi| dHⁿ⁻¹(x). (3.9) Aspbelongs toSBV(Ω), according to theorem2.1, the setS_p is a countable union ofC¹-hypersurfaces. Then, we obtain:

Hⁿ⁻¹(S_p) = sup

A,V

(_∞ X

i=1

Z

Sp∩Ai

|hν_p, ν_ii| dHⁿ⁻¹(x) )

where the supremum is taken over all the families A= (A_i)i=1···∞ of open and pairwise disjoint subsets of Ω andV = (ν_i)i=1···∞ ⊂Sⁿ⁻¹. Applying the inequality (3.9) to each term, we have:

Hⁿ⁻¹(S_p)≤sup

A

(_∞ X

i=1

F−(p, A_i) )

. where the supremum is only taken over all the familiesA.

Since F−(p,·) is super-additive and non decreasing (see (3.1)), then we may conclude that

Hⁿ⁻¹(S_p)≤F−(p,Ω).

3.2 The inequality for the higher Γ-limit (3.5)

To prove the second part (3.5), we construct a sequence of functions (p_k)k∈N

converging top, so that the higher bound of the energy is lower thanF(p,Ω).

We will use the following lemma due to Modica [6].

Lemma 3.2. Let A be an open and bounded subset of Ωwith a non empty Lipschitz boundary∂A. Let

h(x) =

(+dist(x, ∂A) if x∈A,

−dist(x, ∂A) otherwise and for allt >0, S_t={x∈A:h(x) =t}.

Thenh is a Lipschitz function, and we have:

|Dh(x)|= 1 a.e. x ∈Ω and lim

t→0Hⁿ⁻¹(S_t∩Ω) =Hⁿ⁻¹(∂A∩Ω) . We may now prove the second part (3.5) of theorem3.

Proof. Set:

h(x) =

(+dist(x,{p= 0}) if p(x) = 1,

−dist(x,{p= 1}) if p(x) = 0, χ0(t) =

(1 sit≥0, 0 sit <0.

It is clear that p=χ0◦h. We constructp_k asp_k=χ_k◦h. In this case, we have:

Fε_k(p_k) =^Z

Ω 9ε_k|∇χ_k(h(x))|²|∇h(x)|²+χ²_k(h(x))(1−χ_k(h(x)))² ε_k

! dx.

The function p belongs to SBV(Ω). According to theorem 2.1 and lemma 3.2, we have |∇h(x)| = 1 for almost every x ∈ Ω. Then, we may apply coarea formula (see for example [5]) to obtain:

F_εk(p_k) = ^{Z +∞}

t=−∞

ϕ_ε,k(t)Hⁿ⁻¹({h=t}) dt,

where ϕε,k(t) = 9εk|∇χ_k(t)|²+ χ²_k(t)(1−χk(t))²

ε_k .

The minimization of this energy is a 1D-problem. The Euler-lagrange equa- tion associated to this problem is:

(9ε_k(χ⁰_k)²)⁰ = χ²_k(1−χ_k)² ε_k

!⁰

Thus, there existsc_k>0 such that:

9εk(χ⁰_k)² =ck+χ²_k(1−χk)²

ε_k (3.10)

Letχ_k a solution of the equation (3.10) such that:











χ_k(t) = 0 ift≤0,

3√

ε_kχ_k⁰ = r

c_k+^χk2(1−χ_εk ^k)2 ift∈[0, η_k], χ_k(t) = 1 ift≥η_k. Asχ_k⁰ ≥

r c_k

9ε_k, we haveη_k≤ r c_k

9ε_k. If c_k

ε_k →0 thenχ_k◦h converges top almost everywhere.

SetA= 3√

ε_kχ_k⁰ and B = ^{χk(1−χ√}_εk^k). We have A² =c_k+B² and A²+B² = 2AB+c_k+ 2B²−2AB,

= 2AB+ck+ 2B(B−A),

= 2AB+c_k+ 2B(B−^qc_k+B²),

= 2AB+c_k−2c_k B

B+^pck+B². Replacing it inFε_k(p_k) leads to:

Fε_k(p_k) = ^{Z ηk}

t=0(A²+B²)Hⁿ⁻¹({h=t}) dt

= ^{Z ηk}

t=0(2AB+c_k−2c_k B

B+^pc_k+B²)Hⁿ⁻¹({h=t}) dt

≤ Z ηk

t=0(2AB+c_k)Hⁿ⁻¹({h=t}) dt. ReplacingA and B by its values, we get:

Fε_k(p_k)≤ Z ηk

t=06χ_k⁰χ_k(1−χ_k)Hⁿ⁻¹({h=t}) dt +^{Z ηk}

t=0

c_kHⁿ⁻¹({h=t}) dt. Asp∈SBV. So,t→ Hⁿ⁻¹({h=t}) is continuous att= 0 and converges to Hⁿ⁻¹(Sp) with theorem2.1. Thus, the second term converges to 0. Applying Fatou lemma gives

lim supF_εk(p_k) ≤ Hⁿ⁻¹(S_p) lim sup^{Z ηk}

t=0

3

4χ_k⁰(1−χ²_k) dt

≤ Hⁿ⁻¹(S_p)^{Z 1}

0 6s(1−s) ds

≤ Hⁿ⁻¹(Sp).

4 Convergence of solutions

In this section we prove an existence result for the minimizing problem ofEε, withε >0 fixed; we setp_εsuch solution. Then, we show the convergence a.e.

of an extracted subsequence (p_εk)εk∈N, with ε_k →0⁺, to a binary function p∈SBV(Ω). To demonstrate that, we use the Γ-convergence result proved in section3.

4.1 Lower semi-continuity of E_ε

Forε >0 fixed, we can find a minimizer ofEε on W_b^1,2(Ω). Contrary toE, the functionalE_ε is lower semi-continuous for the usual topology.

Theorem 4.1. Let ε > 0, the problem (P_ε) (1.2) admits solutions. More precisely, if (p_k)k∈N ⊂W_b^1,2(Ω) is a minimizing sequence of E_ε, then there exists a sub-sequence(pkl)l∈N converging a.e. to p∈W^1,2(Ω)such that

Eε(p) = lim

l→∞Eε(p_kl) = minⁿEε(q): q∈W_b^1,2(Ω)^o.

Proof. Let (pk)k∈N⊂W_b^1,2(Ω) be a minimizing sequence ofEε. In particular, the sequence (E_ε(p_k))k∈N is bounded. Since

9εβk∇p_kk²_L2(Ω)≤Eε(pk) and

kp_kk²_L2(Ω)≤ kp_kk_L^∞_(Ω)Lⁿ(Ω)≤ Lⁿ(Ω),

then this sequence is bounded in W^1,2(Ω). So, there exists a subsequence (p_kl)l∈N which weakly converges to p ∈ W^1,2(Ω). The functional q → R

Ω|∇q|² is lower semi-continuous for the weak topology of W^1,2(Ω). Ac- cording to Fatou lemma, the functionals

• q→^R_Ω(q−g)²

• q→^R_Ωq²(1−q)²

are lower semi-continuous for the topology of the a.e. convergence. Then, we may conclude

E_ε(p)≤lim inf

l→∞ E_ε(p_kl). Since (p_k)k∈Nis a minimizing sequence then

Eε(p) = lim

l→∞Eε(p_kl) = minⁿEε(q): q∈W_b^1,2(Ω)^o.

4.2 Compactness result of (P_ε)_ε>0

According to4.1, there existsp_εsolution of (P_ε). For numerical applications, we need the convergence of the solutions (p_εk)k∈N for any ε_k → 0⁺ to a solution of (P). In this section, we prove that this family is compact in the following sense.

Theorem 4.2. Let(ε_k)k∈Nbe a sequence converging to0⁺and let(p_k)k∈N⊂ W_b^1,2(Ω) be such that for any k∈N, p_k is a minimizer of E_εk. Then, there exists a subsequence (p_kl)l∈N which converges a.e. to p ∈ B(Ω). Moreover, we have:

i) p(x)∈ {0; 1} a.e. x∈Ω,

ii) p∈SBV(Ω)and Hⁿ⁻¹(S_p)<+∞, iii) p is a solution of (P).

This result also prove that the spaceSBV(Ω;{0; 1}) is "optimal" for the previous Γ-convergence result: it is the largest domain for which this result is satisfied.

Proof. Let q ∈ SBV(Ω) an arbitrary function such that q takes its values in{0; 1}and Hⁿ⁻¹(S_q)<∞. According theorem3, there exists a sequence (q_k)k∈N such that (Eεk(q_k))k∈N converges to E(q). In particular, the se- quence (E_εk(q_k))k∈Nis bounded, we denoteM its higher bound. Sincep_k is a minimizer ofE_εk for anyk then (E_εk(p_k))k∈N is bounded byM too.

We setck= 3p²_k−2p³_k. Note thatck∈W^1,2(Ω) and that the chain rule gives

∇c_k = 6∇p_kp_k(1−p_k), so we have Z

Ω

|∇c_k| = 6^Z

Ω

|∇p_k|p_k(1−p_k)dx,

≤ Z

Ω9ε_k|∇p_k|²dx+^Z

Ω

p²_k(1−p_k)² ε_k dx,

≤ M β .

Thus (c_k)k∈Nis bounded inW^1,1(Ω). So, there exists a subsequence (c_kl)l∈N

which converges a.e. to c ∈ B(Ω). According that x → 3x² −2x³ is a bicontinuous isomorphism from [0,1] to itself, there exists p ∈ B(Ω) such that (p_kl)l∈N converges a.e. top. We have

∀k∈N, Z

Ω

p²_k(1−pk)²

ε_k dx≤ M β . As (ε_k)k∈N converges to 0⁺, Fatou lemma yields

Z

Ω

p²(1−p)²dx= 0,

so thatp(x)∈ {0; 1} a.e. x∈Ω.

Let us proveii. As

∀k∈N, Eε_k(p_k)≤M.

thenF−(p,Ω)≤M <+∞.

Letν ∈Sⁿ⁻¹ be a fixed vector, and (r_k)k∈N⊂W_b^1,2(Ω) such that ( r_k^a.e.→ p,

Fεk(rk,Ω)→F−(p,Ω).

Sincerk∈W^1,2(Ω), then (rk)x (defined by2.1) belongs toW^1,2(Ωx) and we have:

h∇r_k(x+tν), νi=∇(r_k)x. As

Z

Ω 9ε_k|h∇r_k, νi|²+r_k²(1−r_k)² ε_k

!

dx≤F_εk(r_k,Ω). Fubini theorem gives

Z

Ων

Z

Ωx

"

9ε_k|h∇r_k(x+tν), νi|²+r²_k(1−r_k)² εk

#

dtdHⁿ⁻¹(x)≤F_εk(r_k,Ω).

and _Z

Ων

F¹_k((r_k)x,Ωx)dHⁿ⁻¹(x)≤F_εk(r_k,Ω). We apply Fatou lemma to get

Z

Ων

lim infF¹_k((r_k)x,Ωx)dHⁿ⁻¹(x) ≤ F−(p,Ω), Z

Ων

F₋¹((px),Ωx)dHⁿ⁻¹(x) ≤ F−(p,Ω).

We deduce thatF₋¹((px),Ωx)<+∞ is finiteHⁿ⁻¹-a.e. x∈Ων. Otherwise, ((r_k)x)k∈N converges to p_x a.e. on Ωx and p_x takes its values on {0; 1} Hⁿ⁻¹-a.e. x∈Ων. According to lemma3.1, we have

p_x∈SBV(Ωx)

for Hⁿ⁻¹-a.e. x ∈ Ων. Moreover, we have proved that H⁰(Spx ∩Ωx) ≤ F₋¹(p_x,Ωx) and ∇p_x= 0, so we have

Z

Ων

Z

Ωx

|∇p_x|dt +H⁰(S_px)dHⁿ⁻¹(x)<+∞.

Now, the conditions of theorem 2.3 are satisfied. We can conclude that p∈SBV(Ω) and Hⁿ⁻¹(Sp)<+∞.

Let us proveiii.

5 Conclusion

We have shown in section 3 that (Eε)ε Γ-converges to E (theorem 3.1) in SBV and this is the best we can have (theorem4.2). Furthermore, we have shown that this approximation is suitable for numerical experiments since the solutions of (P_ε) are approximations of (P) (theorem 4.1).

Practically, we replace the problem

(P): Min{E(p):p∈SBV(Ω), and p takes its values in{0,1}}

by the problem

(P_ε): MinⁿEε(p): p∈W_b^1,2(Ω)^o

withεfixed and small. In the companion paper [4], we study this approxi- mate problem with the specifics constraints for thin tubes.

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