VARIATIONAL APPROXIMATION OF INTERFACE ENERGIES AND APPLICATIONS

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VARIATIONAL APPROXIMATION OF INTERFACE ENERGIES AND APPLICATIONS

Samuel Amstutz, Daniel Gourion, Mohammed Zabiba

To cite this version:

Samuel Amstutz, Daniel Gourion, Mohammed Zabiba. VARIATIONAL APPROXIMATION OF

INTERFACE ENERGIES AND APPLICATIONS. Interfaces and Free Boundaries, European Math-

ematical Society, 2021, Interfaces and Free Boundaries, pp.59-102. �hal-02390079v2�

APPLICATIONS

SAMUEL AMSTUTZ, DANIEL GOURION, AND MOHAMMED ZABIBA

Abstract. Minimal partition problems consist in finding a partition of a domain into a given number of components in order to minimize a geometric criterion. In applicative fields such as image processing or continuum mechanics, it is standard to incorporate in this objective an interface energy that accounts for the lengths of the interfaces between components. The present work is focused on the theoretical and numerical treatment of minimal partition problems with such interface energies. The considered approach is based on a Γ-convergence approximation combined with convex analysis techniques.

1. Introduction

Consider a partition of a bounded domain Ω of R^d into relatively closed subsets Ω1, . . . ,ΩN, called phases, that may intersect only through their boundaries:

Ω =

N

[

j=1

Ωj, with Ωi∩Ωj =∂Ωi∩∂Ωj∩Ω fori6=j.

Denote the interface separating Ωi and Ωj by Γij :

Γ_ij =∂Ω_i∩∂Ω_j∩Ω fori6=j,

with the additional convention Γ_ii =∅, see Figure 1. The prototype problem of minimal partition can be written as

minimize

N

X

i=1

Z

Ωi

gi(x)dx+I(Ω1, . . . ,ΩN) (1.1) over all partitions (Ω1, . . . ,ΩN) of Ω, whereg1, . . . , gNare given functions inL¹(Ω), andI(Ω1, . . . ,ΩN) is the total interface energy. This energy is here chosen as

I(Ω₁, . . . ,Ω_N) =1 2

X

1≤i<j≤N

α_ijH^d−1(Γ_ij), (1.2)

whereαij≥0 is a coefficient called surface tension associated with Γij andH^d−1(Γij) is thed−1 dimensional Hausdorff measure of Γij. It is convenient to assume that the surface tensions satisfy αij=αjiwheneveri6=j andαii= 0. We denote

SN =

(αij)∈R^N×N :αij =αji andαii = 0 .

In order to guarantee the lower semicontinuity of the interface energy, it is required that the surface tensions be nonnegative and satisfy the triangle inequality [4]

α_ij≤α_ik+α_kj ∀i, j, k. (1.3)

(A1,A2,A3) Universit´e d’Avignon, Laboratoire de Math´ematiques d’Avignon, 301 rue Baruch de Spinoza, 84916 Avignon, France

(A1)Ecole Polytechnique, CMAP, route de Saclay, 91128 Palaiseau, France

E-mail addresses:samuel.amstutz@polytechnique.edu, daniel.gourion@univ-avignon.fr, mohammed.zabiba@univ-avignon.fr.

1

Ω1

Ω2

Ω3

Ω Γ23

Γ₁₃ Γ12

Figure 1. A partition of a domain into sets (Ωj) that intersect only at their boundaries. Interface Γij separates Ωi from Ωj.

This condition is also discussed in [15, 16, 19]. We will therefore mainly place ourselves in the classes of surface tensions

S_N⁺={(αij)∈SN :αij ≥0}, T_N =

(α_ij)∈S_N⁺:α_ij ≤α_ik+α_kj ∀i, j, k .

For the rigorous mathematical analysis, the lower semicontinuity of (1.2) needs to be formulated in an appropriate framework, namely the space of sets of finite perimeter, or Caccioppoli sets [6,9,23].

In this setting, the total interface energy writes 1

2 X

1≤i<j≤N

αijH^d−1(∂MΩi∩∂MΩj∩Ω), (1.4) where Ω1, . . . ,ΩN are now assumed to be sets of finite perimeter in Ω such that Ω =∪^N_i=1Ωi up to a Lebesgue negligible set and |Ωi∩Ωj| = 0 for all i 6= j, denoting by | · | thed-dimensional Lebesgue measure. Moreover,∂MΩi is the measure theoretical (or essential) boundary of Ωiin Ω.

We refer to [6, 9, 23] for details on sets of finite perimeter and geometric measure theory.

Domain functionals of perimetric type are known to be difficult to handle within numerical optimization procedures. The most direct approach in shape optimization relies on the concept of shape derivative, often implemented by means of level-sets, see e.g. the seminal paper [3]

and [2] for a multiphase application. Drawbacks of this setting are that it does not allow all types of topology changes, and that it raises the difficulty, when perimetric terms are involved, of the numerical evaluation of curvatures. In this paper we follow another path, and propose an approximation of the energy (1.4) by a Γ-converging parameterized functional. This latter is constructed upon the solutions of auxiliary elliptic boundary value problems, in the spirit of [7, 8]. This is in contrast with the celebrated Modica-Mortola Γ-convergence approximation of the perimeter [26] which, borrowing the terminology of numerical schemes, could be qualified as explicit. The Modica-Mortola functional, special case of the Ginzburg Landau free energy, has been used in particular by several authors to address minimal partition problems, see e.g. [10–12, 27], and specifically [15] where the energy (1.4) is considered. Closely related to our approach is the parabolic approximation, applied to (1.4) in [19], see also [1, 25] for the two phase case. Nonlocal functionals, either elliptic or parabolic, lend themselves to optimization procedures which are less sensitive to the spatial discretization than local, explicit ones. In particular, descent steps are unrelated to mesh size. As we will see, the elliptic framework has a further advantage: it provides a variational formulation which enables the implementation of alternating minimization schemes.

The separated subproblems may be linear or quadratic and be solved in one shot without line search. Finally, different from Γ-convergence based methods, we mention the convex approximation of minimal partition problems from [17].

On the mathematical side, two main questions are addressed in the present work. The first one is the Γ-convergence of the approximating functionals, which we establish under two alternative sets of assumptions. In the first setting we assume that the surface tensions satisfy an algebraic property, denoted by (αij) ∈ B_N⁺, implying that the total interface energy can be written as a conical combination of perimeters of clusters of phases. This allows to use results on the two-phase

case from [7,8], under generalized forms. In the second setting we only assume that (α_ij)∈T_N, but we suppose that Ω is a Cartesian product of intervals. To prove the Γ−lim inf inequality we follow a very different approach, to a large extent inspired from [19]. Our most novel contribution deals with the second issue, namely the construction of the aforementioned variational formulation.

It is based on Legendre-Fenchel duality arguments, therefore it involves convexity assumptions.

We actually propose two complementary formulations in order to cope with all surface tensions (αij)∈TN.

The paper is organized as follows. In section 2 we recall and extend some useful results from [7,8].

In section 3 we introduce our interface energy approximation and analyze its pointwise convergence.

Section 4 deals with the lower semicontinuity and equicoercivity properties. In section 5 we recall and complement known combinatorial issues concerning the decomposition of the interface energy as a weighted sum of perimeters, and prove our two Γ-convergence results. Sections 6 and 7 are dedicated to the variational formulation. The resulting algorithm is presented in section 8, together with some numerical examples. In section 9 we describe an enrichment of the algorithm in order to take into account volume constraints. A technical lemma is deferred in appendix.

2. Preliminary: a gradient-free perimeter approximation

To set up the mathematical framework, we assume that the hold-all Ω is an open and bounded subset ofR^d,d∈ {2,3}, with Lipschitz boundary, and we first define the functionalF :L^∞(Ω,{0,1})→ R∪ {+∞}by

F(u) =





 1

2|Du|(Ω) ifu∈BV(Ω,{0,1}),

+∞ otherwise.

We recall that the total variation of u satisfies |Du|(Ω) = H^d−1(∂MΩ1 ∩Ω) whenever u ∈ BV(Ω,{0,1}) and uis the characteristic function of a Lebesgue-measurable subset Ω1 of Ω, de- noted by u=χΩ₁, see again [6, 9, 23]. Then we say that Ω1 is a set of finite perimeter and that H^d−1(∂MΩ1∩Ω) is the relative perimeter of Ω1 in Ω. We also define the extended functional ˜F over the convex setL^∞(Ω,[0,1]) by

F˜(u) =

(F(u) ifu∈L^∞(Ω,{0,1}), +∞ otherwise.

In all what follows we denote

hu, vi= Z

Ω

u(x)·v(x)dx

for every pair of scalar or vector valued functionsu, vhaving suitable regularity. It is shown in [7,8]

that a variational approximation of ˜F, in the sense of Γ-convergence, is provided by the family of functionals ( ˜Fε)ε>0 defined by

F˜ε(u) = inf

υ∈H¹(Ω)

εk∇υk²_L2(Ω)+1 ε

kυk²_L2(Ω)+hu,1−2υi

. (2.1)

We recall below some results proven in [7, 8]. The first one is a straightforward reformulation of (2.1) with the help of Euler-Lagrange equations.

Proposition 2.1. Let u∈L²(Ω) be given andLεu:=vε∈H¹(Ω) be the (weak) solution of (−ε²∆v_ε+v_ε =u in Ω,

∂nvε = 0 on∂Ω. (2.2)

Then we have

F˜ε(u) = 1

εh1−Lεu, ui.

It follows straightforwardly from (2.2) thatL_ε1 = 1. Also, the weak formulation yields for all u, v∈L²(Ω)

hLεu, vi= Z

Ω

ε²∇(Lεu)· ∇(Lεv) + (Lεu)·(Lεv) dx,

whereby the operatorL_ε:L²(Ω)→L²(Ω) is self-adjoint and positive definite. It follows that F˜ε(u) =1

εhLεu,1−ui.

Moreover, the weak maximum principle yields 0≤u≤1⇒0≤Lεu≤1.

The second result will be useful for existence issues atεfixed.

Lemma 2.2. The functionalF˜_ε is continuous onL^∞(Ω,[0,1]) for the weak-∗ topology of L^∞(Ω).

The third result establishes the lim inf inequality of the Γ−convergence of the approximating functionals. From now on, convergence statements forε→0 will refer to the convergence of the corresponding quantity considering any sequence (εk) of positive numbers such that lim_k→+∞εk = 0.

Proposition 2.3. Let u∈L^∞(Ω,[0,1]) and(u^ε) be a sequence of functions of L^∞(Ω,[0,1]) such thatu^ε→ustrongly in L¹(Ω). Then we have

lim inf

ε→0

F˜ε(u^ε)≥F˜(u).

Proofs of the lim sup inequality of the Γ-convergence may involve more geometrical aspects, with a possible influence of the space dimension and the shape of Ω. In [8] it was proved for a Lipschitz domain Ω in any dimension, with the help of a recovery sequence (u^ε). However, recovery sequences become problematic in the multiphase case, since independent recovery sequences (u^ε_i)1≤i≤N have no reason to satisfyPN

i=1u^ε_i = 1, even if this property is verified at the limit. In [7] the lim sup inequality was proved for the constant recovery sequenceu^ε=u, which is obviously a remedy to the above limitation, in two dimensions for Ω rectangular. Here we extend this result to Lipschitz domains in dimensiond∈ {2,3}.

Proposition 2.4. For allu∈BV(Ω,{0,1})and allε >0 we have lim sup

ε→0

F˜_ε(u)≤ 1

2|Du|(Ω).

Proof. We first note that for allu∈BV(Ω,{0,1}) we can write F˜ε(u) =Fε(u) := 1

εhu−Lεu, ui.

Moreover, standard arguments provide the variational formulation F_ε(u) = inf

w∈H¹(Ω)

εk∇wk²_L2(Ω)+1

εkw−uk²_L2(Ω)

. We will estimateFε(u) through three steps.

Step 1. In the first step we assume thatu∈H¹(Ω,[0,1]). We have in particular for allw∈ C²(Ω) Fε(u)≤εk∇wk²_L2(Ω)+1

εkw−uk²_L2(Ω), which rewrites

Fε(u)≤ε Z

∂Ω

∂nw(w−u)ds+ε Z

Ω

∇w· ∇udx+1 ε

Z

Ω

(−ε²∆w+w−u)(w−u)dx. (2.3) Here we have used the Green formula forBV functions [6, 9], which applies in Lipschitz domains.

We recall thatuadmits a trace in L¹_Hd−1(∂Ω), and that this trace can be lifted by a function in W₁¹( ˜Ω\Ω), where ˜Ω is a bounded open smooth set containing Ω, see [6, 20]. Call ˜uthe obtained extension ofu, further extended by 0 outside ˜Ω. Inequality (2.3) extends by density to any function w∈ C¹( ˜Ω) with ∆w∈L²( ˜Ω). We choose w=wε:= Φε?u˜ where Φεis the fundamental solution

of the operator −ε²∆ +I. By construction, it holds −ε²∆w_ε+w_ε= ˜ua.e. in ˜Ω. Hence, since

˜

u∈W^1,1( ˜Ω),wε∈ C¹( ˜Ω) [21] and F_ε(u)≤ε

Z

∂Ω

∂_nw_ε(w_ε−u)ds+ε Z

Ω

∇wε· ∇udx. (2.4) The construction from [20] permits to assume the 0≤u˜≤1 a.e. in ˜Ω. Using Φε(x) =ε^−dΦ1(ε⁻¹x) we obtain

ε∇wε(x) = Z

R^d

∇Φ1(z)˜u(x−εz)dz.

Letν be a unit vector of R^d. We infer ε∇wε(x)·ν=

Z

R^d

∇Φ1(z)·νu(x˜ −εz)dz≤ Z

R^d

max(∇Φ1(z)·ν,0)dz.

Due to the radial symmetry of Φ₁we can without loss of generality assume thatν is oriented along the first basis vector ofR^d. It follows that

ε∇wε(x)·ν≤ Z

R^d

max(∂_x1Φ₁(z),0)dz.

We subsequently infer

ε∇wε(x)·ν ≤ Z

R

max Z

R^d−1

∂x1Φ1(z1,z)d¯¯ z,0

dz1

because, due to radial symmetry, the sign of ∂x₁Φ1(z1,z) only depends on the coordinate¯ z1. By uniqueness, the function

z₁7→

Z

R^d−1

Φ₁(z₁,z)d¯¯ z is the one dimensional fundamental solution, i.e.,

Z

R^d−1

Φ1(z1,z)d¯¯ z= 1 2e^−|z1|. We arrive at

ε∇wε(x)·ν ≤ Z +∞

0

1

2e^−z1dz₁=1 2, whereby, sinceν is arbitrary,

ε|∇w_ε(x)| ≤ 1 2. Coming back to (2.4) we obtain

F_ε(u)≤1 2

Z

∂Ω

|w_ε−u|ds+1 2

Z

Ω

|∇u|dx= 1 2 Z

∂Ω

|Φ_ε?u˜−u|ds+1 2

Z

Ω

|∇u|dx. (2.5) Step 2. We now assume thatu∈BV(Ω,[0,1]). By density ofC^∞(Ω) inBV(Ω) for the intermediate convergence [6, 9], there exists a sequence of functions uk ∈ H¹(Ω) such that uk → u in L¹(Ω) and |Duk|(Ω) → |Du|(Ω). The construction by mollifiers allows to assume that 0 ≤ uk ≤ 1.

By Parseval’s equality, as the Fourier transform of Φ1 is FΦ1(ξ) = 1/(1 +|ξ|²), we infer that Φ₁∈L²(R^d). Since obviouslyu_k →ualso inL²(Ω), ˜u_k →u˜ inL²(R^d) by continuity of the trace operator for the intermediate convergence, and F_ε is continuous on L²(Ω), taking limits in (2.5) yields

Fε(u)≤ 1 2

Z

∂Ω

|Φε?u˜−u|ds+1 2 Z

Ω

|Du|. (2.6)

Step 3. It remains to estimate the first integral in (2.6), which denoting wε = Φε?u˜ can be rewritten

Z

∂Ω

|wε−u|ds= Z

∂Ω

Z

R^d

Φ1(y) (˜u(x−εy)−u(x))˜ dy

ds(x).

Letα >0. Due to the decay of Φ₁ at infinity there existsρ >0 such that Z

∂Ω

Z

R^d\Bρ(0)

Φ1(y) (˜u(x−εy)−u(x))˜ dy

ds(x)≤α.

The Cauchy-Schwarz inequality yields Z

∂Ω

|wε−u|ds≤ H^d−1(∂Ω)^1/2kΦ1k_L2(R^d)

Z

∂Ω

Z

Bρ(0)

|˜u(x−εy)−u(x)|˜ dyds(x)

!1/2

+α.

By a change of variable this rewrites as Z

∂Ω

|wε−u|ds≤ H^d−1(∂Ω)^1/2kΦ1k_L2(R^d)

Z

∂Ω

ε^−d Z

Bερ(0)

|˜u(x−z)−u(x)|˜ dzds(x)

!1/2

+α.

Theorem 3.87 of [6] states the following: forH^d−1- a.e. x∈∂Ω it holds

t→0limt^−d Z

Ω∩B_t(x)

|u(y)−u(x)|dy= 0.

Obviously the same limit holds for the exterior part, which entails

ε→0limε^−dρ^−d Z

Bερ(0)

|˜u(x−z)−˜u(x)|dz= 0

forH^d−1- a.e. x∈∂Ω. Then it follows from Lebesgue’s dominated convergence theorem that

ε→0lim Z

∂Ω

ε^−d Z

Bερ(0)

|u(x˜ −z)−u(x)|˜ dzds(x) = 0.

We infer that, forεsmall enough, Z

∂Ω

|wε−u|ds≤2α.

This completes the proof.

As straightforward consequences of Propositions 2.3 and 2.4, one obtains the desired Γ-convergence and pointwise convergence results.

Theorem 2.5. When ε → 0, the functionals F˜_ε Γ−converge in L^∞(Ω,[0,1]) endowed with the strong topology ofL¹(Ω) to the functionalF˜ defined by

F(u) =˜





 1

2|Du|(Ω) ifu∈BV(Ω,{0,1}),

+∞ otherwise.

Theorem 2.6. For allu∈L^∞(Ω,[0,1]) it holds

ε→0lim

F˜ε(u) = ˜F(u). (2.7)

3. Approximation of interface energies: pointwise convergence

Given two subsets Ωi and Ωj of Ω, we look for an approximation of the interface energy H¹(∂MΩi∩∂MΩj∩Ω). The starting point is the following result established within the proof of Proposition 1 of [5].

Lemma 3.1. Let Ωi,Ωj be sets of finite perimeter such that |Ωi∩Ωj| = 0. There exists an H^d−1-negligible set Lsuch that

∂M(Ωi∪Ωj)\L⊂∂MΩi∆∂MΩj ⊂∂M(Ωi∪Ωj).

We obtain the following extension of Proposition 1 of [5].

Proposition 3.2. Let (Ω_i)_i=1,...mbe sets of finite perimeter such that|Ωi∩Ω_j|= 0for anyi6=j.

Then

H^d−1(∂M(∪^m_i=1Ωi)∩Ω) =

m

X

i=1

H^d−1(∂MΩi∩Ω)−2 X

1≤i<j≤m

H^d−1(∂MΩi∩∂MΩj∩Ω).

Proof. First we derive from Lemma 3.1 that if|Ωi∩Ω_j|= 0 then

H^d−1(∂_M(Ω_i∪Ω_j)∩Ω) =H^d−1(∂_MΩ_i∩Ω) +H^d−1(∂_MΩ_j∩Ω)−2H^d−1(∂_MΩ_i∩∂_MΩ_j∩Ω). (3.1) This proves the proposition form= 2. The general case is obtained by induction. For readability we present the proof form= 3. Using (3.1) we obtain

H^d−1(∂M(Ω1∪Ω2∪Ω3)∩Ω) =H^d−1(∂MΩ1∩Ω) +H^d−1(∂MΩ2∩Ω) +H^d−1(∂MΩ3∩Ω)

−2H^d−1(∂_MΩ₁∩∂_MΩ₂∩Ω)−2H^d−1(∂_M(Ω₁∪Ω₂)∩∂_MΩ₃∩Ω).

Using Lemma 3.1 we get

H^d−1(∂_M(Ω₁∪Ω₂)∩∂_MΩ₃∩Ω) =H^d−1((∂_MΩ₁∩∂_MΩ₃∩Ω)∆(∂_MΩ₂∩∂_MΩ₃∩Ω)).

Now, we will prove that

H^d−1(∂MΩ1∩∂MΩ2∩∂MΩ3) = 0. (3.2) Call Ω_i¹² the set of points of density ¹₂ relatively to Ω_i, see e.g. [6]. By definition we have

Ω₁¹² ∩Ω₂¹² ∩Ω₃¹² =∅.

As a consequence, it follows that

0 =H^d−1(Ω₁¹² ∩Ω₂¹² ∩Ω₃¹²) =H^d−1(∂_MΩ₁∩∂_MΩ₂∩∂_MΩ₃), (3.3) since the two sets above coincide up to anH^d−1-negligible set, see [6]. We infer that

H^d−1((∂MΩ1∩∂MΩ3∩Ω)∆(∂MΩ2∩∂MΩ3∩Ω))

=H^d−1(∂_MΩ₁∩∂_MΩ₃∩Ω) +H^d−1(∂_MΩ₂∩∂_MΩ₃∩Ω) and subsequently

H^d−1(∂M(Ω1∪Ω2∪Ω3)∩Ω) =H^d−1(∂MΩ1∩Ω) +H^d−1(∂MΩ2∩Ω) +H^d−1(∂MΩ3∩Ω)

−2H^d−1(∂MΩ1∩∂MΩ2∩Ω)−2H^d−1(∂MΩ1∩∂MΩ3∩Ω)−2H^d−1(∂MΩ2∩∂MΩ3∩Ω).

This proves the result form= 3.

We have now all the ingredients to prove the pointwise convergence result.

Theorem 3.3. Let Ωi,Ωj be two subsets of finite perimeter of Ω such that |Ωi ∩Ωj| = 0. If ui=χΩi anduj=χΩj, then

H^d−1(∂MΩi∩∂MΩj∩Ω) = lim

ε→0

2

εhLεui, uji.

Proof. By Proposition 3.2, we have H^d−1(∂MΩi∩∂MΩj∩Ω) = 1

2

H^d−1(∂MΩi∩Ω) +H^d−1(∂MΩj∩Ω)− H^d−1(∂M(Ωi∪Ωj)∩Ω) . Using Theorem 2.6 we obtain

H^d−1(∂MΩi∩∂MΩj∩Ω) = lim

ε→0

hF˜ε(ui) + ˜Fε(uj)−F˜ε(ui+uj)i

= lim

ε→0

1

εh1−L_εu_i, u_ii+1

εh1−L_εu_j, u_ji −1

εh1−L_ε(u_i+u_j), u_i+u_ji

= lim

ε→0

2

εhL_εu_i, u_ji.

Figure 2. (left) given partition, (right) convergence history ofG_ε(u_i, u_j) com- puted with the FEM (solid lines), the FDM (dashed lines) and the exact values (horizontal lines).

We denote

Gε(ui, uj) = 1

εhLεui, uji. (3.4)

We present an example to illustrate the pointwise convergence of the functionalGεin Figure 2. The values of the functionG_εare computed using two discretization methods, namely the finite element method (FEM) with Q1 elements and the finite difference method (FDM) with 5 points stencil.

The parameterεhas the dimension of a length. In fact, in view of (2.2), it is a characteristic width of the diffuse interface represented by the slow variable v_ε. Thus we start with a characteristic size of Ω, namelyε₀=ε_max = max(m, n) where (m, n) is the size of the grid (its stepsize is fixed as unitary). Then we divideεby two between each computation, that is, we chooseεi=εmax/2ⁱ. In order to approximate (2.2) properly, ε must not be taken significantly smaller than the grid resolution. Thus we stop the algorithm as soon asεi ≤εmin= 1. We observe that the computed values ofGε(ui, uj) are always smaller using the FDM than using the FEM. This is due to higher diffusion of the FEM.

4. Lower semicontinuity and equicoercivity

4.1. Lower semicontinuity. The following important result is found in [4]. An alternative proof is given in [19] when Ω is a Cartesian product of intervals, in which case it is also a consequence of Theorem 5.11.

Theorem 4.1. Let (αij)∈S_N⁺. The condition (1.3) is necessary and sufficient for the function I: (Ω1, . . . ,ΩN)7→ 1

2 X

1≤i<j≤N

αijH^d−1(∂MΩi∩∂MΩj∩Ω)

to be lower semicontinous for the convergence in measure in the set ofN-tuples (Ω1, . . . ,ΩN) of Lebesgue-measurable subsets ofΩsuch thatχ_Ωi ∈BV(Ω) for alli andPN

i=1χ_Ωi= 1.

This property will lead to the existence of minimizers for the exact minimal partition problem in Theorem 5.2. In addition, lower-semicontinuity is a necessary condition for Γ-convergence [9, 14], which will be addressed later. Equicoercivity is another important property. Basically, together with Γ-convergence, it implies that sequences of minimizers of approximating functionals converge up to a subsequence to a minimizer of the limiting functional, see again, e.g., [9, 14].

4.2. Equicoercivity. We will rely on the following theorem from [7].

Theorem 4.2. Let u^ε be a sequence of functions of L^∞(Ω,[0,1]) such that sup_ε>0F˜_ε(u^ε)<+∞.

There existsu∈L^∞(Ω,{0,1})such thatu^ε→ustrongly in L¹(Ω)for a subsequence.

We set

Iε(u1, . . . , uN) = X

1≤i<j≤N

αijGε(ui, uj) =1 ε

X

1≤i<j≤N

αijhLεui, uji.

We now prove the equicoercivity of the functionalsI_ε.

Theorem 4.3. Assume that (α_ij) ∈ S_N⁺ with α_ij ≥ α > 0. Let (u^ε₁, . . . , u^ε_N) be a sequence of N-tuples of functions inL^∞(Ω,[0,1])such thatPN

i=1u^ε_i = 1for allεandsupε>0Iε(u^ε₁, . . . , u^ε_N)<

+∞. For all i, there exists ui ∈L^∞(Ω,{0,1}) such that u^ε_i →ui strongly inL¹(Ω) for a subse- quence. Moreover we havePN

i=1ui= 1.

Proof. Using (3.4), we obtain X

1≤i<j≤N

αijGε(u^ε_i, u^ε_j) = 1 ε

X

1≤i<j≤N

αijhLεu^ε_i, u^ε_ji

≥ α ε

X

1≤i<j≤N

hL_εu^ε_i, u^ε_ji= α 2ε

X

1≤i6=j≤N

hL_εu^ε_i, u^ε_ji= α 2ε

N

X

i=1

hL_εu^ε_i,

N

X

j=1 j6=i

u^ε_ji.

Due toP

j6=iu^ε_j = 1−u^ε_i we infer X

1≤i<j≤N

αijGε(u^ε_i, u^ε_j)≥ α 2

N

X

i=1

1

εhLεu^ε_i,1−u^ε_ii= α 2

N

X

i=1

F˜ε(u^ε_i).

The result follows from Theorem 4.2.

5. Conical combinations of perimeters and Γ-convergence

In this section we rewrite the interface energy as a linear combination of perimeters of aggregated phases. If all coefficients can be taken nonnegative (conical combination) then the Γ-convergence of the approximating functional is straightforward. Therefore special attention is paid to the signs of the coefficients.

LetS⊂ {1, . . . , N}. From now on, we will denote ΩS =∪i∈SΩi and ¯S={1, . . . , N} \S.

5.1. Algebraic properties of interface energies.

Lemma 5.1. Let Ω1, . . . ,ΩN be subsets of finite perimeter of Ω such that|Ω\ ∪^N_i=1Ωi| = 0 and

|Ω_i∩Ω_j|= 0 fori6=j. LetL_ij =H^d−1(∂_MΩ_i∩∂_MΩ_j∩Ω), P_S =H^d−1(∂_MΩ_S∩Ω). Then PS=X

i∈S j /∈S

Lij =P_S.

Proof. By the definition of the essential boundary, we have

∂MΩi=∂M R^d\Ωi

=∂M

j6=i∪Ωj∪ R^d\Ω

.

As an elementary property of the essential boundary, we have that ∂M(A∪B) ⊂∂MA∪∂MB.

Moreover, as Ω is open, we have∂MΩ∩Ω =∅. This yields

∂M

j6=i∪Ωj∪ R^d\Ω

∩Ω⊂

j6=i∪∂MΩj

∩Ω,

which implies that

∂MΩi∩Ω = ∪

j6=i(∂MΩj∩∂MΩi∩Ω). (5.1) Fori6=j, i6=k, j6=k, following (3.2), we have

H^d−1((∂MΩi∩∂MΩj∩Ω)∩(∂MΩi∩∂MΩk∩Ω)) =H^d−1(∂MΩi∩∂MΩj∩∂MΩk∩Ω) = 0.

(5.2) We deduce from (5.1),(5.2) that

P_{i}=X

j6=i

L_ij.

From this fact and Proposition 3.2, we obtain that PS =X

i∈S

P_{i}−2 X

(i,j)∈S² i<j

Lij =X

i∈S j /∈S

Lij.

We arrive at the announced existence result.

Theorem 5.2. Assume that(αij)∈TN withαij≥α >0. Letg1,· · ·, gN ∈L¹(Ω). The problem minimizeJ(Ω₁, . . . ,Ω_N) :=

N

X

i=1

Z

Ω_i

g_i(x)dx+I(Ω1, . . . ,Ω_N), (5.3) in the set of N-tuples (Ω1, . . . ,ΩN) of Lebesgue-measurable subsets of Ωsuch that χΩ_i ∈BV(Ω) for alliand PN

i=1χΩ_i= 1, admits at least a solution.

Proof. We have the inequality J(Ω1, . . . ,ΩN)≥ −

N

X

i=1

kgik_L1(Ω)+α 2

X

1≤i<j≤N

H^d−1(∂MΩi∩∂MΩj∩Ω).

Lemma 5.1 entails

J(Ω1, . . . ,ΩN)≥ −

N

X

i=1

kgikL¹(Ω)+α 4

N

X

i=1

H^d−1(∂MΩi∩Ω).

Therefore, for a minimizing sequence (Ω^k₁, . . . ,Ω^k_N), the quantityPN

i=1H^d−1(∂MΩ^k_i∩Ω) is bounded.

By a standard property of bounded sequences of sets of finite perimeters, see e.g. [6, 9, 23], there exists a family (Ω₁, . . . ,Ω_N) of subsets of finite perimeters of Ω such that Ω^k_i → Ω_i in measure for each i, for a non-relabeled subsequence. Equivalently, χ_Ωk

i → χ_Ωi in L¹(Ω), which implies thatPN

i=1χΩ_i = 1. The lower-semicontinuity of Theorem 4.1 shows that (Ω1, . . . ,ΩN) is a global

minimizer.

5.2. Algebraic properties of approximate interface energies. We now prove the approxi- mate counterpart of Lemma 5.1.

Lemma 5.3. Let Ωi, . . . ,ΩN be subsets of finite perimeter of Ω such that |Ω\ ∪^N_i=1Ωi| = 0 and

|Ω_i∩Ω_j|= 0fori6=j. Letu_i=χ_Ωi,L^ε_ij = 1

εhL_εu_i, u_ji,P^ε_S= 1

εh1−L_εP

i∈Su_i,P

i∈Su_ii. Then P^ε_S =X

i∈S j /∈S

L^ε_ij =P^ε_S.

Proof. We have

P^ε_S =1 ε

* 1−Lε

X

i∈S

ui,X

i∈S

ui

+

= 1 ε

* Lε

X

i∈S

ui,1−X

i∈S

ui

+ . Using 1−P

i∈Sui=P

j /∈Suj we obtain P^ε_S= 1

ε

* Lε

X

i∈S

ui,X

j /∈S

uj

+

= 1 ε

X

i∈S j /∈S

hLεui, uji=X

i∈S j /∈S

L^ε_ij.

We emphasize that the properties stated in Lemmas 5.1 and 5.3 are formally the same. This will allow to obtain similar reformulations for the interface energy and its approximation. The approximate counterpart of Theorem 5.2 is stated below.

Theorem 5.4. Assume that (α_ij)∈S_N⁺ withα_ij≥α >0. Letg₁,· · ·, g_N ∈L¹(Ω). The problem minimizeJ_ε(u₁, . . . , u_N) :=

N

X

i=1

Z

Ω

u_ig_i(x)dx+I_ε(u₁, . . . , u_N), (5.4) in the set of N-tuples(u1, . . . , uN)∈L^∞(Ω,[0,1])^N such that PN

i=1ui= 1a.e., admits at least a solution.

Proof. Consider a minimizing sequence (u^k₁, . . . , u^k_N) ∈ L^∞(Ω,[0,1])^N such that PN

i=1u^k_i = 1.

Up to a subsequence, this sequence converges weakly-∗ to some (u1, . . . , uN) ∈ L^∞(Ω,[0,1])^N. Obviously it holds PN

i=1ui = 1. By Lemma 2.2, (u1, . . . , uN) is a minimizer of (5.4).

5.3. Matrix representation of algebraic properties. We define the column vector L made of the values (Lij) in a chosen order. Similarly we define the column vector α of the surface tensions (αij) and P the vector gathering the values PS, for S ∈ S ⊂ P({1, . . . , N}). The set S is made as small as possible by exploiting the property of complementation. We adopt the following construction: whenN is odd the elements ofS are the subsets of{1, . . . , N}containing between 1 and (N−1)/2 elements; whenN is even the elements ofS are the subsets of{1, . . . , N} containing between 1 and N/2−1 elements and the subsets containing N/2 elements including 1 (see Table 1 for N ≤5). An alternative - bijective - set could be taken as the set of nonempty subsets of {1, . . . , N−1}, so that]S = 2^N−1−1. In view of Lemma 5.1, we can define a matrix M= (mij)∈R^]S×(^N2) such that

P=ML. (5.5)

Note thatmij ∈ {0,1}.

Letβ= (βS)S∈S. Starting from the dot products

β·P=β·ML=M^|β·L, one infers that

X

1≤i<j≤N

α_ijL_ij =X

S∈S

β_SP_S (5.6)

holds for anyLand correspondingPas soon as the columns of coefficients satisfy the linear system

M^|β=α. (5.7)

Due to

Lij= 1

2(Pi+Pj−Pij), (5.8)

N=2 {{1}}

N=3 {{1},{2},{3}}

N=4 {{1},{2},{3},{4},{12},{13},{14}}

N=5 {{1},{2},{3},{4},{5},{12},{13},{14},{15}, {23},{24},{25},{34},{35},{45}}

Table 1. The setS of values ofS.

it turns out that Mhas full rank. However there are in general multiple ways to find a β corre- sponding to a givenα. For the purpose of proving a Γ-convergence property, possible nonnegative solutions of this system will be privileged.

5.4. Existence of conical combination. We define the set

B_N⁺ ={(αij)∈S_N :∃(βS)≥0 s.t.α=M^|β} ⊂S_N⁺.

We now address the identification of the set B⁺_N. We consider S_N, T_N and B⁺_N as subsets of the Euclidean space R^N(N−1)/2. Note that S_N is the full linear space, while T_N and B_N⁺ are polyhedral convex cones. Indeed, T_N is defined as intersection of half-spaces of R^N(N−1)/2, and B_N⁺ is the convex cone generated by the row vectors of M. The setsT_N and B_N⁺ are sometimes called the semimetric cone (or metric cone) and the cut cone (or Hamming cone), respectively, see for example [18]. For the sake of completeness, we recall that a matrix (α_ij)∈ S_N is called `¹- embeddable if there exists some integerKandNpointsx1, . . . , xN ∈R^Ksuch thatαij =kxi−xjk1

for all 1≤i < j≤N. It is known that the set of`¹-embeddable matrices is equal toB_N⁺ (see for example [18], proposition 4.2.2). It is also known thatB⁺_N ⊂TN for anyN ≥2 and thatB_N⁺ =TN

forN ≤4. Nevertheless we present our own proofs of these results, without using the concept of

`¹-embeddability.

Theorem 5.5. For anyN ≥2 it holdsB_N⁺ ⊂TN.

Proof. Using the conic descriptions of B_N⁺ and TN, we only have to check that any row vector of Mis an element of TN. Consider an arbitrary row of M. It corresponds to a setS ∈ S. Call (m_ij) the entries of this row vector in the system of indices associated with phases. Recall that m_ij ∈ {0,1}. Consider a nontrivial triangle inequalitym_ij ≤m_ik+m_kj (withi, j and kdistinct integers) defining T_N. In view of Lemma 5.1, if both i and j are in S, then m_ij = 0 and the inequality is satisfied. The same holds if both i and j are not in S. If i ∈ S and j /∈ S, then m_ij = 1. In this case, eitherk∈S andm_kj= 1, ork /∈S andm_ik= 1. In both cases the triangle inequality is satisfied. Obviously the same occurs if i /∈S andj ∈ S. Thus any row vector ofM

belongs toTN, which implies thatB_N⁺ ⊂TN.

Theorem 5.6. If N= 3,4 thenTN ⊂B⁺_N.

Proof. We will use the notationsβi forβ_{i} andβij forβ_{ij}. We treat separately the two cases.

• Case 1: N = 3. The unique solution of (5.7) is β₁= −α₂₃+α₁₂+α₁₃

2 , β₂= −α₁₃+α₁₂+α₂₃

2 , β₁₂= −α₁₂+α₁₃+α₂₃

2 .

If (α_ij)∈T₃, thenβ₁, β₂,andβ₁₂are nonnegative, which implies thatT₃⊂B₃⁺.

• Case 2: N = 4. Solving (5.7) forN = 4, we choose the particular solution β12= −α12+α₁₄+α₂₄

2 , β13= −α13+α₁₄+α₃₄

2 , β23= −α23+α₂₄+α₃₄

2 ,

β1= α12+α13−α24−α34

2 , β2= α12+α23−α14−α34

2 ,

β3= α13+α23−α14−α24

2 , β4= 0.

It is immediate to see that if (α_ij) ∈ T₄, then β₁₂, β₁₃, and β₂₃ are nonnegative. Now, we want to prove that if (α_ij) ∈ T₄, then β₁, β₂, and β₃ are nonnegative too. Let us define for i = 1, . . . ,4, Σ_i = P

j6=iα_ij. Up to reordering the phases, we assume that Σ4≤Σ3≤Σ2≤Σ1. Then we have







Σ₄≤Σ₁ Σ4≤Σ2

Σ₄≤Σ₃

⇒







α₁₄+α₂₄+α₃₄≤α₁₂+α₁₃+α₁₄ α14+α24+α34≤α12+α23+α24

α₁₄+α₂₄+α₃₄≤α₁₃+α₂₃+α₃₄

⇒







α24+α34≤α12+α13

α₁₄+α₃₄≤α₁₂+α₂₃ α14+α24≤α13+α23

⇒





 β1≥0 β₂≥0 β3≥0

.

We now discuss the numerical search for some (βS)≥ 0, given coefficients (αij). Let us first recall the definition of a conical combination and Carath´eodory’s theorem. Given a finite number of vectorsv1, v2, . . . , vpin a real vector space, a conical combination of these vectors is a vector of the form

λ1v1+λ2v2+. . .+λpvp, where the real numbers λ₁,· · ·, λ_p are non-negative.

Theorem 5.7 (Carath´eodory). In a vector space of dimensionn, all conical combination of m vectors(m > n), can be written as a conical combination of nof these vectors.

By the above theorem and the linear system (5.7), when α = (αij) ∈ B_N⁺, α can be written as a conical combination of ^N₂

rows of the matrix M. Calling B the corresponding submatrix, we have Bβ =α, with β ≥0. Denoting k = rankB, then αbelongs to the space spanned by k linearly independent columns ofB. By Carath´eodory’s theorem again, αis a conical combination of kcolumns ofB. By completion of these vectors (since M^|has full rank), αwrites as a conical combination of ^N₂

linearly independent columns ofM^|. This leads to Algorithm 1.

Data: Givenα= (αij)∈R(^N2)^×1, M∈R^]S×(^N2).

1 repeat

2 Loop on the set of square invertible submatrices Λ∈R(^N₂)^×(^N₂) ofM^|;

3 Computeβ= Λ⁻¹α;

4 until β ≥0;

5 Completeβ by zeros at the entries corresponding to the columns ofM^|that have been removed.

Algorithm 1:Search for (β_S)≥0.

Using Algorithm 1 we are in particular able to find counterexamples to Theorem 5.6 when N = 5. For instance, it is immediately seen that the matrix

(αij) =













0 2 3 2 1

2 0 1 2 3

3 1 0 3 3

2 2 3 0 1

1 3 3 1 0













satisfies the triangle inequality, but Algorithm 1 terminates without finding anyβ ≥0.

The complexity of Algorithm 1 rapidly grows with N: for example for N = 6, there are 300540195 ^N₂

× ^N₂

submatrices of M^|. Hence, it is impossible in practice to use this algo- rithm forN >5. For this reason we propose a second algorithm. Let (B⁺_N)^◦denote the polar cone

ofB⁺_N. Moreau’s decomposition theorem directly implies thatα∈B⁺_N if and only if the projection ofαon (B_N⁺)^◦ is 0. This leads us to the following positive definite quadratic program:

min

a_i·y≤0∀ikα−yk², (5.9)

where (a_i) are the rows ofM. Thenα ∈B⁺_N if and only if the optimal value of this program is 0. This program is easily tractable up toN = 13. This technique, however, does not provide any possibleβ. Note that the algorithms developed in sections 8 and 9 do not require the knowledge of suchβ.

5.5. Γ-convergence with nonnegative coefficients. Define the set E˜N =

(

(u1, . . . , uN)∈L^∞(Ω,[0,1])^N :

N

X

i=1

ui= 1 a.e.

)

and the functional ˜I: ˜EN →Rsuch that I˜(u1, . . . , uN) =





 1 2

P

1≤i<j≤Nα_ijH¹(∂_MΩ_i∩∂_MΩ_j∩Ω) ifu_i∈BV(Ω,{0,1})∀i, ui=χ_Ωi,

+∞ otherwise.

Theorem 5.8. If (αij)∈B_N⁺, then the functionalsIε Γ- converge to I˜ in E˜N endowed with the strong topology ofL¹(Ω)^N.

Proof. We first prove the lim inf inequality. Let (u^ε_i)∈E˜N be a sequence such that (u^ε_i) converges tou_i. From (3.4), (5.6) and Lemma 5.3 we have

X

1≤i<j≤N

αijGε(u^ε_i, u^ε_j) =1 ε

X

S∈S

βS

* 1−Lε

X

i∈S

u^ε_i,X

i∈S

u^ε_i +

. (5.10)

This entails lim inf

ε→0

X

1≤i<j≤N

αijGε(u^ε_i, u^ε_j)≥X

S∈S

lim inf

ε→0

1 εβS

* 1−Lε

X

i∈S

u^ε_i,X

i∈S

u^ε_i +

.

We infer from Theorem 2.5 and (5.6) that lim inf

ε→0

X

1≤i<j≤N

αijGε(u^ε_i, u^ε_j)≥I˜(u1, . . . , uN).

Second, due to the pointwise convergence (Theorem 3.3), the lim sup inequality holds for the

constant recovery sequence.

5.6. Γ-convergence in the general case. Here we generalize Theorem 5.8 to arbitrary surface tensions when Ω is a Cartesian product of intervals. Our proof is widely inspired from [19], nevertheless it incorporates some adaptations to our context. In [19] the functional

Eε(u1,· · ·, uN) = 1 ε

X

1≤i<j≤N

αij

Z

D

Gε? uiujdx (5.11) is considered. The convolution kernelG_ε is mainly chosen as the Gaussian, and more generally it is assumed to fulfill some properties which are not all satisfied in our case. The main ingredient in the proof of [19] is an approximate monotonicity argument. Here we follow the same path. We start with a rough estimate of derivative, however different from [19] since we exploit here the underlying boundary value problems instead of the convolution structure. Therefore, we do not need at this stage any geometric assumption on Ω.