A simple phase-field approximation of the Steiner problem in dimension two

A simple phase-field approximation of the Steiner problem in dimension two

A. Chambolle B. Merlet L. Ferrari

In this paper we consider the branched transportation problem in 2D asso- ciated with a cost per unit length of the form 1 +αm where m denotes the amount of transported mass and α >0 is a fixed parameter (notice that the limit case α= 0 corresponds to the classical Steiner problem). Motivated by the numerical approximation of this problem, we introduce a family of func- tionals ({F_ε}_ε>0) which approximate the above branched transport energy.

We justify rigorously the approximation by establishing the equicoercivity and the Γ-convergence of {Fε} as ε ↓ 0. Our functionals are modeled on the Ambrosio-Tortorelli functional and are easy to optimize in practice. We present numerical evidences of the efficiency of the method.

1 Introduction

In this paper, we introduce a phase-field approximation of a branched transportation energy for lines in the plane. Our main goal is to derive a computationally tractable approximation of the Steiner problem (of minimizing the length of lines connecting a given set of points) in a phase-field setting. Similar results have recently be obtained by [BLS15], however we believe our approach is slightly simpler and numerically easier to implement. We show that we can modify classical approximations for free discontinuity problems [MM77, AT90, Iur13, CFI15] to address our specific problem, where the limiting energy is concentrated only on a singular one-dimensional network (and roughly measures its length). Numerical results illustrate the behaviour of these elliptic approximations. In this first study, we limit ourselves to the two-dimensional case, as in that case lines can locally be seen as discontinuities of piecewise constant functions, so that our construction is deriving in a quite simple ways from the above mentioned previous works on free discontinuity problems. Higher dimension is more challenging, from the topological point of view; an extension of this approach is currently in preparation.

We now introduce precisely our mathematical framework. Let Ω ⊂ R² be a convex, bounded open set. We consider measures σ ∈ M(Ω,R²) that write

σ =θξ· H¹

x

^M,

where M is a 1-dimensional rectifiable set orientated by a Borel measurable mapping ξ :M →S¹ andθ :M →R₊is a Borel measurable function representing the multiplicity.

Such measure is called a rectifiable measure. We follow the notation of [OS11] and write σ = U(M, θ, ξ). Given a cost function f ∈ C(R₊,R₊), we introduce the functional defined on M(Ω,R²)→R₊∪ {+∞} as

E_f(σ) :=





 Z

M

f(θ) dH¹ if σ =U(θ, ξ, M), +∞ in the other cases.

Given a sequence of N + 1 distinct points S = (x₀, . . . , x_N) ∈ Ω^N+1, we consider the minimization of E_f(σ) for σ ∈ M(Ω,R²) satisfying the constraint

∇ ·σ =N δ_x0 −

N

X

i=1

δ_xi inD⁰(R²). (1.1)

The distributional support of such σ connects the source inx₀ to the sinks in x₁,· · ·, x_N. In general, a model for branched transport connecting a set of sources to a set of sinks (represented by two discrete probabilistic measures supported on a set of points in Ω) is obtained by choosingf(θ) =|θ|^αwith 0< α <1 and minimizing the associated functional under a divergence constraint similar to equation (1.1). The direct numerical optimization of the functionalE_f is not easy because we do not knowa priorithe topological properties of the tree M. For this reason it is interesting to optimize an “approximate” functional defined on more flexible objects such as functions. Such approximate model has been introduced in [OS11] where the authors study the Γ-convergence (see [Bra02]) of a family of functionals inspired by the well known work of Modica and Mortola [MM77]. Another effort in this direction can be found in the work [BLS15] where is studied an approximation to the Steiner Minimal Tree problem ([GP68], [AT04] and [PS13]) by means of analogous techniques.

Here, we consider variational approximations of some energies of the form E_f through a family of functionals modeled on the Ambrosio-Tortorelli functional [AT90]. For being more precise, we need to introduce some material. Let ρ ∈ C_c^∞(R²,R₊) be a classical radial mollifier with suppρ⊂B₁(0) andR

ρ= 1. Forε∈(0,1], we setρ_ε(x) =ε⁻²ρ(ε⁻¹x) and we define the space V_ε(Ω) of square integrable vector fields whose weak divergence satisfy the constraint

∇ ·σ_ε = N δ_x0 −

N

X

j=1

δ_xj

!

∗ρ_ε. (1.2)

Forη =η(ε)>0, we note W_ε(Ω) =

φ∈H¹(Ω) : η ≤φ≤1 in Ω, φ≡1 on ∂Ω . Then we define the energy F_ε:M(Ω,R²)×L¹(Ω)→[0,+∞] as

F_ε(σ, φ) :=





 Z

Ω

1

2εφ²|σ|² dx+ Z

Ω

ε

2|∇φ|²+ (1−φ)²

2ε dx, if (σ, φ)∈V_ε(Ω)×W_ε(Ω),

+∞ in the other cases.

(1.3) The first integral in the definition of the energy will be refered to as the “constraint com- ponent” while the second integral will be regarded as the “Modica-Mortola component”.

Let us briefly describe the qualitative properties of the associated minimization problem.

First notice that the constraint (1.2) enforces σ to be non zero on a set connecting S.

Next, the constraint component of the energy strongly penalizes φ²|σ|² so that φ should be small in the region where |σ| is large. On the other hand the behavior of φ is con- trolled by the Modica Mortola component that forces φ to be close to 1 away from a one-dimensional set as ε converges to 0. As a consequence, we expect the support of σ and the energy to concentrate on a one dimensional set connecting S. The main part of the paper consists in making rigorous and quantitative this analysis.

From now on, we assume that there exists someα ≥0 such that η

ε

−→ε↓0 α. (1.4)

We note M_S(Ω) the set of R²-valued measures σ ∈ M(R²,R²) with support in Ω such that the constaint (1.1) holds. We define the limit energy Eα : M(Ω,R²)×L¹(Ω) → [0,+∞] as

E_α(σ, φ) =





 Z

M

(1 +α θ) dH¹ if φ≡1, σ∈ M_S(Ω) and σ =U(M, θ, ξ), +∞ in the other cases.

(1.5) We prove the Γ-convergence of the sequence (F_ε) to the energyE_α asε↓0. More precisely the convergence holds in M(Ω,R²)×L¹(Ω) where M(Ω,R²) is endowed with the weak star topology and L¹(Ω) is endowed with its classical strong topology.

We establish the following lower bound.

Theorem 1.1 (Γ−lim inf). For any sequence (σε, φε) ⊂ M(Ω,R²)×L¹(Ω) such that σ_ε * σ^∗ and φ_ε →φ in the L¹(Ω) topology, with (σ, φ)∈ M(Ω,R²)×L¹(Ω)

lim inf

k→+∞ F_ε(σ_ε, φ_ε)≥ E_α(σ, φ).

In this statement and throughout the paper, we make a small abuse of language by noting (a_ε)ε∈(0,1] and calling sequence a family {a_ε} labeled by a continuous parameter ε∈(0,1]. In the same spirit, we call subsequence of (a_ε), any sequence (a_εj) withε_j →0 as j →+∞.

To complete the Γ-convergence analysis, we establish the matching upper bound.

Theorem 1.2 (Γ−lim sup). For any(σ, φ)⊂ M(Ω,R²)×L¹(Ω) there exists a sequence (σ_ε, φ_ε) such that σ_ε * σ^∗ and φ_ε→φ in the L¹(Ω) topology and

lim sup

k→+∞

F_ε(σ_ε, φ_ε)≤ E_α(σ, φ).

Moreover, under the assumption α > 0 we prove the equicoercivity of the sequence (F_ε).

Theorem 1.3 (Equicoercivity). Assume α > 0. For any sequence (σ_ε, φ_ε)_ε∈(0,1] ⊂ M(Ω,R²)×L¹(Ω) with uniformly bounded energies, i.e.

sup

ε

F_ε(σ_ε, φ_ε) < +∞,

there exist a subsequence ε_j ↓ 0 and a measure σ ∈ M_S(Ω,R²) such that σ_εj → σ with respect to the weak-∗ convergence of measures and φ_εj → 1 in L¹(Ω). Moreover, σ is a rectifiable measure (i.e. it is of the form σ =U(M, θ, ξ)).

Remark 1. Observe that letting α = 0 in equation (1.5) we obtain E₀(σ) = H¹({x ∈ M : θ(x)>0}) where σ=U(M, θ, ξ). This is exactly the functional associated with the Steiner Minimal Tree problem. Unfortunately, the hypothesis α > 0 is necessary in the compactness Theorem 1.3.

The fact that we are working in dimension 2 is fundamental for the proof of Theorem 1.1 as it allows to locally rewrite the vector field σ_ε as the rotated gradient of a function.

Structure of the paper: In Section 2 we introduce some notation and several tools and notions on SBV functions and vector field measures. In Section 3 we study a first family of energies obtained by substituting ∇u forσ in the definition ofF_ε. In Section 4 we prove the equicoercivity result, Theorem 1.3 and we establish the lower bound stated in Theorem 1.1. In Section 5 we prove the upper bound of Theorem 1.2. Finally, in the last section, we present and discuss various numerical simulations.

2 Notation and Preliminary Results

In the following Ω⊂⊂Ωˆ ⊂R^dare bounded open convex sets. GivenX ⊂R^d(in practice X = Ω or X = ˆΩ), we denote by A(X) the class of all open subsets of X and byA_S(X) the subclass of all simply connected open sets O ⊂ X such that O∩S =∅. We denote by (e₁, . . . , e_d) the canonical orthonormal basis of R^d, by| · | the euclidean norm and by h·,·i) the euclidean scalar product inR^d. The open ball of radiusr centered at x∈R^d is denoted by Br(x). The (d−1)-dimensional Hausdorff measure inR^d is denoted by H^d−1. We write |E| to denote the Lebesgue measure of a measurable set E ⊂ R^d. When µ is a Borel meaure and E ⊂ R^d is a Borel set, we denote by µ

x

^E the measure defined as µ

x

^{E(F) =µ(E∩F).}

Let us remark that from Section 4 onwards, we work in dimension d= 2.

ForO ∈ A(Ω), the functional F_ε(·,·;O) is the functional obtained by substutingO for Ω in the definition of Fε (see (1.3)). Similarly we define the local version Eα(·,·;O) of Eα.

2.1 BV(Ω) functions and Slicing

BV(Ω) is the space of functions u ∈ L¹(Ω) having as distributional derivative Du a measure with finite total variation. For u∈BV(Ω), we denote by S_u the complement of the Lebesgue set of u, that isx6∈S_u if and only if

lim

ρ→0⁺

1

|B_ρ(x)|

Z

Bρ(x)

|u(y)−z|dy= 0

for some z ∈R. We say thatx is an approximate jump point of u if there exist ξ ∈S^d−1 and distinct pointsa, b∈R such that

limρ↓0

1

|B_ρ⁺(x, ξ)|

Z

Bρ⁺(x,ξ)

|u(y)−a|dy= 0 and lim

ρ↓0

1

|B_ρ⁻(x, ξ)|

Z

B⁺ρ(x,ξ)

|u(y)−b|dy= 0, where B_ρ^±(x, ξ) := {y ∈ Bρ(x) : ±hy−x, ξi ≥ 0}. Up to a permutation of a and b and a change of sign of ξ, this characterizes the triplet (a, b, ξ) which is then denoted by (u⁺, u⁻, ν_u). The set of approximate jump points is denoted byJ_u. The following theorem holds [AFP00].

Theorem 2.1. The set S_u is countably H^d−1-rectifiable and H^d−1(S_u\J_u) = 0. Moreover Du

x

^{Ju = (u+−u−)νuHd−1}

x

^{Ju and}

T an^d−1(J_u, x) =ν_u(x)^⊥ for H^d−1-a.e. x∈Ju.

We write the Radon-Nikodym decomposition of Du as Du = ∇u dx+D^su. Setting D^cu=D^su

x

^(Ω\Su) we get the decomposition

Du=∇udx+ (u⁺−u⁻)ν_uH^d−1

x

^Ju+Dcu.

Moreover the Cantor part is such that if H^d−1(E) <∞, we have |D^cu|(E) = 0 [AFP00, Thm. 3.92]. In particular, we have the following useful consequence

H^d−1(E) = 0 =⇒ |Du|(E) = 0. (2.1)

We frequently use the notation [u] for the jump function (u⁺−u⁻) :J_u →R.

When d= 1 we use the symbol u⁰ in place of ∇uand u(x^±) to indicate the right and left limits of u atx.

Let us introduce the space of special functions of bounded variation and a variant:

SBV(Ω) :={u∈BV(Ω) :D^cu= 0},

GSBV(Ω) :={u∈L¹(Ω) : max(−T,min(u, T))∈SBV(Ω)∀T ∈R}.

Eventually, in Section 3, the following space of piecewise constant functions will be useful.

P(Ω) ={u∈GSBV(Ω) : ∇u= 0}.

To conclude this section we recall the slicing method for functions with bounded variation.

Letξ ∈S^d−1 and let

Π_ξ:={y∈R^d:hy, ξi= 0}.

If y∈Πξ and E ∈R^d, we define the one dimensional slice E_ξ,y :={t∈R:y+tξ ∈E}.

Foru: Ω→R, we define u_ξ,y : Ω_ξ,y →R as

u_ξ,y(t) :=u(y+tξ), t∈Ω_ξ,y.

Functions inGSBV(Ω) can be characterized by one-dimensional slices (see [Bra98, Thm. 4.1]) Theorem 2.2. Let u∈GSBV(Ω). Then for all ξ ∈S^d−1 we have

u_ξ,y ∈GSBV(Ω_ξ,y) for H^d−1-a.e. y∈Π_ξ. Moreover for such y, we have

u⁰_ξ,y(t) = h∇u(y+tξ), ξi for a.e. t ∈Ω_ξ,y, J_uξ,y ={t ∈R:y+tξ ∈J_u},

and

u_ξ,y(t^±) =u^±(y+tξ) or u_ξ,y(t^±) = u^∓(y+tξ)

according to whether hν_u, ξi>0 or hν_u, ξi<0. Finally, for every Borel function g : Ω→ R,

Z

Πξ

X

t∈J_uξ,y

g_ξ,y(t) dH^d−1(y) = Z

Ju

g|hν_u, ξi|dH^d−1. (2.2) Conversely if u ∈L¹(Ω) and if for all ξ ∈ {e₁, . . . , e_d} and almost every y ∈Π_ξ we have u_ξ,y ∈SBV(Ω_ξ,y) and

Z

Πξ

|Du_ξ,y|(Ω_ξ,y) dH^d−1(y)<+∞

then u∈SBV(Ω).

2.2 Rectifiable vector Measures

Let us introduce the linear operator⊥that associates to each vector v = (v₁, v₂)∈R² the vector v^⊥ = (−v₂, v₁) obtained via a 90^◦ counterclockwise rotation of v. Notice that the

⊥ operator maps divergence free R²-valued measures onto curl free R²-valued measures.

Let O ⊂R² be a simply connected and bounded open set. By Stokes Theorem, for any divergence free measureσ∈ M(O,R²) there exists a functionu∈BV(Ω) with zero mean value such that σ =Du^⊥. On the other hand for u∈ P(Ω) σ :=Du^⊥ is divergence free and by Theorem 2.1, σ= (u⁺−u⁻)ν_u^⊥H¹ =U(J_u,[u], ν_u^⊥).

Let us now produce an elementary example of measureγ of the form U(M, θ, ξ).

Example 1. Given two pointsx, y ∈Ω we consider the smooth path from xtoy defined as

r(t) :=x+t y−x

|x−y| for t∈[0,|x−y|].

We define the measure γ ∈ M(Ω,R²) by (φ, γ) :=

Z

hφ(r(t)),r(t)i˙ dt for any φ∈ C(Ω,R²).

We then have γ =U([x, y],1, ξ) with ξ= _|y−x|^y−x. Notice that ∇ ·γ =δ_x−δ_y.

Similarly we obtain a measure satisfying this property by substituting forr any Lipschitz path fromx to y.

We will make use of the following construction.

Lemma 2.1. Given S = (x₀,· · · , x_N)∈Ω^N+1 a sequence of N + 1 distinct points, there exist a vector measure γ = U(Mγ, θγ, ξγ) and a finite partition (Ωi) ⊂ A(Ω) of Ω such that

a) ∇ ·γ =−N δ_x0 +PN i=1δ_xi, b) θ_γ :M_γ → {1, N},

c) each Ω_i is a polyhedron,

d) M_γ ⊂S

i∂Ω_i,

e) Ω_i is of finite perimeter for each i and Ω_i∩Ω_j =∅ for i6=j, f ) L²(Ω\ ∪_iΩ_i) = 0.

Moreover if M is a 1dimensional countably rectifiable set, we can choose γ and (Ω_i) such that H¹(M ∩S

i∂Ω_i) = 0.

Proof. Let us fix a point p ∈ Ω\S. By Example 1 we can construct a measure γ_i with

∇ ·γ_i =δ_xi−δ_p for i∈ {0,· · · , N}. We define γ =−N γ₀+

N

X

i=1

γ_i.

By construction (a) holds true. Moreover, up to a small displacement ofpwe may assume that [p, x_i]∩[p, x_j] ={p} fori6=j so that (b) holds.

Next, letD_jbe the straight line supporting [p, x_j]. We define the sets (Ω_i) as the connected components of Ω\(D₀∪ · · · ∪D_N). We see that (c, d, e, f) hold true.

For the last statement, we observe that by the coarea formula, we haveH¹(M∩S

i∂Ω_i) = 0 for a.e. choice of p.

Ω

Ω γ

σ

1

2

x3

ν

p

r3

r0

r

r2

Figure 1: Example of the construction of the H¹-rectifiable measure γ (red) and of the partition {Ω_i} (gray) in the case σ (green) is being an H¹-rectifiable vector measure.

3 Local Result

In this section we introduce a localization of the family of functionals (F_ε) (see (1.3)).

We establish a lower bound and a compactness property for these local energies.

Localization. Let O ∈A_S(Ω), for uε∈H¹(O) and φ_ε ∈H¹(O), we define LF_ε(u_ε, φ_ε;O) := F_ε(∇u_ε, φ_ε;O).

Notice that for ε < d(O, S), we have ∇ ·σ_ε ≡0 in O for any σ_ε ∈ V_ε(Ω). By the Stokes theorem we have Du_ε=σ_ε^⊥ for some u∈H¹(O) and we have

F_ε(σ_ε, φ_ε;O) =LF_ε(u_ε, φ_ε;O).

The remaining of the section is devoted to the proof of

Theorem 3.1. Let (u_ε)ε∈(0,1] ⊂H¹(O)be a family of functions with zero mean value and let (φε) ⊂H¹(O) such that φε ∈ H¹(O,[η(ε),1]). Assume that c0 := sup_εLFε(uε, φε;O) is finite, then there exist a subsequence ε_j and a function u∈BV(Ω) such that

a) φ_εj →1 in L²(O),

b) u_εj →u with respect to the weak-∗ convergence in BV, c) u∈ P(O).

Furthermore for any u∈ P(O) and any sequence (u_ε, φ_ε) as above such that u_ε * u, we^∗ have the following lower bound of the energy:

lim inf

ε→0 LFε(uε, φε;O)≥ Z

Ju∩O

[1 +α|[u]|] dH^d−1.

The proof is achieved in several steps and mostly follows ideas from [Iur13] (see also [CFI15]).

In the first step we obtain (a) and (b). In steps 2. we prove(c) and the lower bound for one dimensional slices. Finally in step 3. we prove (c) and the lower bound in dimension d. The construction of a recovery sequence that would complete the Γ-limit analysis is postponed to the global model in Section 5.

Proof. Step 1. Item(a)is a straightforward consequence of the definition of the functional.

Indeed, we have Z

O

(1−φ_ε)² dx ≤εLF_ε(σ_ε, φ_ε) ≤ c₀ε −→^ε↓0 0.

For (b), since (u_ε) has zero mean value, we only need to show that sup_ε{|Du_ε|(O) : k ∈ N}<+∞. Using Cauchy-Schwarz inequality we get

[|Du_ε|(O)]² = Z

O

|∇u_ε| 2

≤

ε Z

O

1 φ²_ε

1 ε

Z

O

φ²_ε|∇u_ε|²

. (3.1)

By assumption, the second therm in the right hand side of (3.1) is bounded by 2c₀. In order to estimate the first term we split O in the two sets {φ_ε < 1/2} and {φ_ε ≥ 1/2}.

We have,

ε Z

O

1 φ_ε = ε

Z

{φ_ε<1/2}

1 φ²_ε +ε

Z

{φ_ε≥1/2}

1 φ²_ε.

Since φ_ε ≥η and (1−t)² is decreasing in the intervalO the following inequalities hold Z

{φ_ε<1/2}

1

φ²_ε ≤ 2ε η²(1−1/2)²

Z

O

(1−φε)²

2ε ≤ 8ε

η

2

c0, Z

{φε≥1/2}

1 φ²_ε ≤

Z

O

1

(1/2)² = 4|O|.

Combining these estimates with (3.1) we obtain [|Du_ε|(O)]² ≤ ε²

η² 16c²₀+ 8ε|O|c₀ −→^ε↓0 16c²₀

α² < ∞. (3.2)

This establishes (b).

Step 2. In this step we suppose O to be an interval of R. We first prove that u is piecewise constant. The idea is that in view of the constraint component of the energy, variations of u_ε are balanced by low values ofφ_ε. On the other hand the Modica-Mortola component of the energy implies that φ_ε'1 in most of the domain and that transitions fromφ_ε '1 toφ_ε '0 have a constant positive cost (and therefore can occur only finitely many times).

Step 2.1. proof of u∈ P(O). Let us define Bε :=

x∈O:φε(x)< 3 4

⊃ Aε :=

x∈O :φε(x)< 1 2

, (3.3)

and let

C_ε ={I connected component of B_ε :I∩A_ε6=∅}. (3.4) Let us show that the cardinality ofCε is bounded by a constant independent ofε. Letεbe fixed and consider an interval I ∈C_ε. Let a, b∈I¯such that {φ_ε(a), φ_ε(b)}={1/2,3/4}.

Using the usual Modica-Mortola trick, we have LF_ε(u_ε, φ_ε;I) ≥

Z

I

ε|φ⁰_ε|²+(1−φ_ε)²

4ε dx≥

Z

(a,b)

|φ⁰_ε|(1−φ_ε) dx ≥ Z 3/4

1/2

(1−t)dt = 3 2⁵. Since all the elements of Cε are disjoint, we deduce from the energy bound that

#C_ε ≤2⁵c₀/3,

where we note #C_ε the cardinality of C_ε. Next, up to extraction we can assume that

#C_ε =N is fixed. The elements of C_ε are written on the form I_i^ε = (m^ε_i −w^ε_i, m^ε_i +w_i^ε) for i= 1,· · · , N with m^ε_i < m^ε_i+1. Sinceφ_ε→1 in L¹(O) we have

X

I_i^ε∈C_ε

|I_i^ε|=X

i

2w_i^ε→0. (3.5)

Up to further extraction, we can assume that each sequence (m^ε_i) converges inO. We call m₁ ≤ m₂ ≤ · · · ≤m_N their limits. We now prove that |Du|(O\ {m_i}^N_i=1) = 0. For this, we fixx∈O\ {m_i}^N_i=0 and establish the existence of a neighborhoodB_δ(x) of xfor which

|Du|(B_δ(x)) = 0. Let 0< δ ≤1/2 min_i|x−m_i|. Equation (3.5) ensures that for ε small enoughB_δ(x)∩C_ε=∅. Notice that from the definitions in (3.3) and (3.4) we have that φ_ε ≥1/2 outsideC_ε. Hence, using Cauchy-Scwarz inequality, we have forε small enough, Z

Bδ(x)

|u⁰_ε|dx 2

≤2δ Z

Bδ(x)

|u⁰_ε|² dx≤(2δ)(2ε)4 1

2ε Z

Bδ(x)

φ_ε|²u⁰_ε|² dx

≤ 16c₀εδ −→^ε↓0 0.

By lower semicontinuity of the total variation on open sets we conclude that|Du|(B_δ(x)) = 0. This establishes u∈ P(O) with J_u ⊂ {m₁,· · · , m_N}.

Step 2.2. Proof of the lower bound. Without loss of generality, we prove the lower bound in the case J_u ={0} and D:=u(0⁺) =−u(0⁻)>0. Using the same argument as

in [Iur13, Pag. 7] for any 0 < d < D there exist six points y₁ < x¹_ε ≤x˜¹_ε <x˜²_ε ≤ x²_ε < y₂ such that

ε→0limφ_ε(y₁) = lim

ε→0φ_ε(y₂) = 1, limε→0φε(x¹_ε) = lim

ε→0φε(x²_ε) = 0, u_ε(˜x¹_ε) = −D+d, u_ε(˜x²_ε) =D−d.

Using the Modica-Mortola trick in the intervals (y₁, x¹_ε) and (x²_ε, y₂) as above, we compute:

lim inf

ε↓0 LFε(uε, φε; (y1, x¹_ε)∪(x²_ε, y2))≥lim inf

ε↓0

Z x¹_ε y1

(1−φε)|φ⁰_ε|dx+

Z y2

x²_ε

(1−φε)|φ⁰_ε|dx ≥ 1.

(3.6) For the estimation on the interval I_ε = (˜x¹_ε,x˜²_ε) let us introduce:

Gε :=

w∈H¹(Iε) :w(˜x¹_ε) = −D+d, w(˜x²_ε) = D−d , Z_ε :=

z ∈H¹(I_ε) :η ≤z ≤1 a.e. on I_ε , H_ε(w, z) :=

Z

Iε

1

2εz²|w⁰|²+(1−z)² 2ε

dx, h_ε(z) = inf

w∈Wε

H_ε(w, z) for z∈Z_ε.

Note that by Cauchy-Schwarz inequality, we have for w∈G_ε and z ∈Z_ε, Z

Iε

z²|w⁰|² ≥ Z

Iε

|w⁰|dx 2Z

Iε

1 z²

−1

≥ 4(D−d)² Z

Iε

1 z²

−1

. We deduce the lower bound

h_ε(z) ≥ 4(D−d)²

2ε Z

Iε

1 z² dx

−1

+ Z

Iε

(1−z)² 2ε

dx. (3.7)

Let us remark that optimizingHε(w, z) with respect tow∈Gεwe see that this inequality is actually an equality.

Consider for 0< λ <1 the inequalities:

Z

{x∈I_ε:φε≥λ}

1

φ²_ε ≤ L¹(Iε)

λ² and

Z

{x∈I_ε:φε<λ}

1

φ²_ε ≤ 1 (1−λ)²

2ε η²

Z

Iε

(1−φε)²

2ε dx

. Applying both of them in (3.7) we obtain

LFε(uε, φε, Iε)≥hε(φε)

≥ 2(D−d)²

εL¹(Iε)

λ² + _(1−λ)¹ 2

2ε² η²

R

Iε

(1−φε)²

2ε dx+

Z

Iε

(1−φ_ε)² 2ε

dx

≥2(1−λ)η

ε(D−d)−(1−λ)²η² 2ε

L¹(I_ε)

λ² (3.8)

where the latter is obtained by minimizing the function:

t 7→ 2(D−d)²

εL¹(Iε)

λ² + _(1−λ)¹ 2

2ε² η² t +t.

Therefore we can pass to the limit in (3.8) and obtain:

lim inf

ε↓0 LFε(uε, φε, Iε)≥(1−λ)α2(D−d).

Sending λ and d to 0 and recalling the estimation in (3.6) we get lim inf

ε↓0 LFε(uε, φε,(y1, y2)) ≥ 1 +α2D = 1 +α|u(0⁺)−u(0⁻)|. (3.9) Step 3. Using Fubini’s decomposition we can rewrite the energy obtaining for every ξ ∈S^d−1,

LF_ε(u_ε, φ_ε;O) ≥ Z

Πξ

Z

O^ξy

1

2ε(φ²_ε)^ξ_y|(u⁰_ε)^ξ_y|²+ ε

2|(φ⁰_ε)^ξ_y|² +(1−(φ_ε)^ξ_y)²

2ε dt

!

dH^d−1(y).

Since LF_ε(u_ε, φ_ε;O) is bounded, forH^d−1 almost every y∈ Π_ξ, the inner integral in the latter is also bounded, furthermore it corresponds to the functional on the one dimensional slice O^ξ_y studied in the previous step evaluated on the couple ((u_ε)^ξ_y,(φ_ε)^ξ_y). Taking the lim inf of the above quantity, using (3.9), we get by Fatou’s lemma that for any ξ ∈S^d−1 and H^d−1 almost every y∈Ω_ξ

Z

Πξ

X

mi∈(J_u)^ξy

1 +α|u^ξ_y(m⁺_i )−u^ξ_y(m⁻_i )|

dH^d−1(y)≤lim inf

ε↓0 LF_ε(u_ε, φ_ε;O).

Therefore in force of Theorem 2.2 we have u ∈ SBV(O). Moreover, since (u⁰)^ξ_y = 0 on each slice, we have u∈ P(O). Applying identity (2.2) we get

lim inf

ε→0 LFε(uε, φε;O)≥ Z

Ju∩O

|νu·ξ|[1 +α|[u]|] dH^d−1. (3.10) In order to conclude, we use the following localization method stated by Braides in [Bra98, Prop. 1.16].

Lemma 3.1. Let µ : A(X) → [0,+∞) be a superadditive set function and let λ be a positive measure on X. For any i ∈ N let ψ_i be a Borel function on X such that µ(A)≥R

Aψ_i dλ for all A∈ A(X). Then µ(A)≥

Z

A

ψ dλ where ψ := sup_iψ_i.

We introduce the superadditive increasing set function µdefined on A(O) by µ(A) := Γ−lim inf

ε→0 LF_ε(u; )), for any A∈ A(O) and we let λ be a Radon measure defined as

λ:= [1 +α|u(x⁺)−u(x⁻)|]H^d−1

x

^Ju.

Fix a sequence (ξ_i)i∈N dense inS^d−1. By (3.10) we have µ(O)≥

Z

O

ψ_i dλ, i∈N, where

ψ_i(x) :=

(|hν_u(x), ξ_ii| if x∈J_u, 0 if x∈O\J_u. Hence by Lemma 3.1 we finally obtain

lim inf

ε→0 LF_ε(u_ε, φ_ε;O)≥ Z

O

sup

i

ψ_i(x) dµ= Z

Ju∩O

[1 +α|[u]|] dH^d−1.

4 Equicoercivity and Γ-liminf

We first prove the compactness property stated in the introduction. Let us consider a sequence (σ_ε, φ_ε)∈ M(Ω,R²) uniformly bounded in energy by c₀ <+∞,

0≤ F_ε(σ_ε, φ_ε)≤c₀ for ε∈(0,1]. (4.1) Proof of Theorem 1.3. First observe that by definition (1.3) and equation (4.1), we have σε ∈Vε(Ω) and φε ∈Wε(Ω).

Next, substituting|σ_ε|for|∇u_ε|in the argument ofStep 1. of the proof of Theorem 3.1, inequality (3.2) reads

|σ_ε|(Ω)≤ s

16ε²

η² c²₀ + 8ε|Ω|c₀ −→^ε↓0 4c0

α < ∞.

Thus the total variation of (σε) is uniformly bounded and there exists σ ∈ MS(Ω) such that up to extraction σ_ε →σ weakly-∗ in M(Ω).

Now, considering the last term in the energy (1.3) we have Z

Ω

(1−φ_ε)² dx≤2εF_ε(σ_ε, φ_ε)≤2ε c₀ →0.

Hence, φ_ε →1 in L²(Ω).

Let us now study the structure of the limit measure σ. Let us recall that ˆΩ is a bounded convex open set such that Ω⊂Ω and let us extendˆ σ_ε by 0 andφ_εby 1 in ˆΩ\Ω.

Obviously we have F_ε(σ_ε, φ_ε; ˆΩ) = F_ε(σ_ε, φ_ε; Ω), therefore for any O ∈ A_s( ˆΩ) applying the localization described in Section 3 we can associate to eachσ_ε a functionu_ε ∈H¹(O) with mean value 0 such that σ_ε = ∇^⊥u_ε in O. By Theorem 3.1 there exists u ∈ P(O) such that up to extraction u_ε* u. Eventually, by uniqueness of the limit, we get^∗

σ

x

^{O =−[u]νJ⊥}uH¹

x

^(Ju∩O).

Since we can cover Ω\S by finitely many sets O ∈ A_s( ˆΩ), this shows that σ decomposes as

σ = U(Mσ, θσ, ξσ) +

N

X

j=0

cjδxj

| {z }

µ

.

By Lemma 2.1 there exists a rectifiable measureγ =U(Mγ, θγ, ξγ) such that∇·(σ+γ) = 0 andH¹(M_γ∩M_σ) = 0. Then there existsu∈BV(Ω) such thatDu=σ^⊥+γ^⊥. From (2.1), we deduce |Du|(S) = 0 which implies |µ|(S) =P

|c_j|= 0. Hence c_j = 0 for j = 0, . . . , N and σ writes in the form U(Mσ, θσ, ξσ).

Let us now use the local results of Section 3 to prove the lower bound.

Proof of Theorem 1.1. Let (σ_ε, φ_ε) as in the statement of the theorem. Without loss of generality, we can suppose thatF_ε(σ_ε, φ_ε)<+∞. Theorem 1.3 then ensures the existence of a rectifiable measure σ =U(M_σ, θ_σ, ξ_σ) with supp(σ)⊂Ω such that σ_ε * σ.^∗

Let ˆΩ be as in the previous proof and let us define µ = Γ −lim inf_εF_ε(σ_ε, φ_ε) and λ = α|σ|+H¹

x

^Mσ. Consider the countable family of sets {O_i} ⊂ A_S( ˆΩ) made of the open rectanglesO_i ⊂Ωˆ\S with vertices in Q² and letψ_i := 1_Oi. The local result stated in Theorem 3.1 gives for any i∈N

µ(A)≥µ(O_i∩A)≥λ(O_i∩A) = Z

A

ψ_i dλ.

Therefore Lemma 3.1 gives Γ−lim inf

ε↓0 F_ε(σ_ε, φ_ε) =µ( ˆΩ)≥λ( ˆΩ) =α|σ|(Ω) +H¹(M_σ) since sup_iψ_i is the constant function 1.

5 Upper bound

5.1 A density result

In order to obtain the upper bound we first provide a density lemma. We show that measures which have support contained in a finite union of segments, are dense in energy.

Without loss of generality let us assume that σ ∈ M_S(Ω) is such that E_α(σ) < ∞. In particular σ =U(M_σ, θ_σ, ξ_σ) is aH¹-rectifiable measure. Applying Lemma 2.1 we obtain aH¹-rectifiable measureγ =U(M_γ, θ_γ, ξ_γ) and a partition of Ω made of polyhedrons{Ω_i} such that M_γ ⊂ ∪_i∂Ω_i,H¹(M_σ∩ ∪_i∂Ω_i) = 0 and σ+γ is divergence free.

From the above properties, we can write

σ^⊥+γ^⊥ =Du

for some u ∈ P(Ω). Our strategy is the following, using existing results [BCG14], we build an approximating sequence foruon each Ω_j whose gradient is supported on a finite union of segments. We then glue these approximations together to obtain a sequence (w_j) approximating u in ˆΩ. The main difficulty is to establish that Dw_j

x

^[∪i∂Ωi] is close to Du

x

^{[∪i∂Ωi] =γ⊥.}

Lemma 5.1 (Approximation ofu). There exists a sequence (w_j)⊂ P( ˆΩ)with the follow- ing properties:

a) w_j →u weakly in BV( ˆΩ), b) suppw_j ⊂Ω,

c) lim sup_j→∞Eα(wj,1)≤ Eα(u,1),

d) J_wj is contained in a finite union of segments for any j ∈N, e) |Dw_j−Du|(∪∂Ω_i)→0.

Proof. Step 1. In order to apply the results of [BCG14], we first need to modify u and the energy. Let us note the energy density functionf(t) = 1 +αtand fork ≥0 andt ≥0 let us introduce the approximation

f_k(t) := min{(2^k/2+α2^−k/2)√

t, f(t)}.

We have 0≤f_k ≤f and f_k ≡f on [2^−k,+∞). Notice thatf_k is continuous, sub-additive

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

f f k1 f k2

Figure 2: Graph of f and two of its approximations f_k1 and f_k2 with k₁ < k₂. and increasing on [0,+∞) and that f_k(0) = 0 with limt→0 fk(t)

t = +∞. We define the associated energy for functions v ∈ P( ˆΩ) as Ef_k(v,Ω) :=ˆ R

Jv∩Ωˆfk([v]) dH¹.

Now we note P_k( ˆΩ) the set of functions v ∈ P( ˆΩ) such that v( ˆΩ) ⊂ 2^−kZ. For these functions we have|v⁺(x)−v⁻(x)| ≥2^−k forH¹-almost every x∈J_v. Consequently, there holds

E_fk(v) = E_f(v).

For each fixedk ≥0, let us introduce the function u_k = 2^−kb2^kuc

wherebtcdenotes the integer part of the real t. Note that u_k∈ P_k( ˆΩ) with J_uk ⊂J_u and ku−u_kk∞≤2^−k. Notice also since

(u⁺_k −u⁻_k)−(u⁺−u⁻)

≤2^−k we have

|Du_k−Du|( ˆΩ) ≤ 2^−kH¹(J_u) (5.1) In particular u_k →u strongly in BV( ˆΩ). Moreover, we see that

E_fk(u_k) = E_f(u_k) ≤ E_f(u) +α2^−kH¹(J_u). (5.2)

Step 2. Let us approximate the function u_k. Let us fix k ≥ 0 and Ω_i. We can apply Lemma 4.1 of [BCG14] to the function u_k

x

^Ωi and to the energy E_fk(·,Ω_i). We obtain a sequence (w_jⁱ) which enjoys the following properties:

w_jⁱ(Ωi)⊂uk(Ωi)⊂2^−kZ, ∀j ∈N, hence w_jⁱ ∈ Pk( ˆΩ), w_i^j →u_k inL¹(Ω_i) as j →+∞,

j→+∞lim E_fk(w_jⁱ,Ω_i) = lim

j→+∞E_f(w_jⁱ,Ω_i) = E_f(u_k,Ω_i), J_w^j

i is contained in a finite union of segments for any j ∈N, Z

∂Ωi

|T w_jⁱ −T uk|dH¹ →0 whereT :BV(Ωi)→L¹(∂Ωi) denotes the trace operator.

Let us now define globally

w_j :=X

j

w_jⁱ1_Ωi. From the above properties, we have w_j * u^∗ _k,

limE_f(w_jⁱ,Ω) =ˆ E_f(u_k,Ω_i) (5.3) and

|Dwj−Duk|(∪i∂Ωi) → 0 asj → ∞. (5.4) Eventually, using a diagonal argument, we have proved the existence of a sequence (w_j)⊂ P( ˆΩ) complying to items (a), (b) and (d) of the lemma. Moreover, item (c) is the consequence of (5.2) and (5.3) and item (e) follows from (5.1) and (5.4).

Going back to theH¹-rectifiable measures σ=U(M_σ, θ_σ, ξ_σ), we define the sequence σ_j :=−Dw_i^⊥−γ.

We recall that γ = U(M_γ, θ_γ, ξ_γ) with M_γ ⊂ ∪∂Ω_i. In particular γ = −Du^⊥

x

^{(∪i∂Ωi).}

We deduce from the previous lemma:

Lemma 5.2. There exists a sequence (σ_j)∈ M_S(Ω) with the properties:

- σ_j →σ with respect to weak-∗ convergence of measures,

- σ_j =U(M_σj, θ_σj, ξ_σj) with M_σj contained in a finite union of segments, - lim sup_j→∞Eα(σj,1)≤ Eα(σ,1).

5.2 Construction of a recovery sequence

Let us prove the Γ-limsup inequality stated in Theorem 1.2. Recall that the latter consists in finding a sequence (σ_ε, φ_ε) for any given couple (σ, φ) ∈ M(Ω,R²)×L¹(Ω) such that σ_ε * σ,^∗ φ_ε→φ inL¹(Ω) and

lim sup

ε↓0

F_ε(σ_ε, φ_ε)≤ E_α(σ, φ). (5.5)

WhenE_α(σ, φ) = +∞the inequality is valid for any sequence therefore by definition (1.5) we can assumeσ =U(M, θ, ξ) andφ= 1. Furthermore by density Lemma 5.2 is sufficient to consider measures of the form

σ=

n

X

i=1

U(M_i, θ_i, ξ_i), (5.6)

where M_i is a segment, θ_i ∈ R₊ is H¹-a.e. constant and ξ_i is an orientation of M_i for each i. Without loss of generality we can suppose that for each couple of segments M_i, M_j, for i 6= j, the intersection M_i∩M_j is at most a point (called branching point) not belonging to the relative interior of M_i and M_j. We firstly produce the estimate (5.5) for σ composed by a single segment thus let us assume σ =θe₁· H¹

x

^{(0, l)× {0}.}

Notation: Let us fix the values a_ε:=





 θα ε

2 if α >0 ε if α= 0

, b_ε :=εln

1−η ε

and r_ε = max{ε, a_ε}.

Let d∞(x, S) be the distance function from x to the set S ⊂ Ω relative to the infinity norm on R² and Q_r(P) = {x ∈ R² : d_∞(x, P) ≤ r} the square centered in P of size 2r and sides parallel to the axes. Introduce the sets

I_aε :={x∈R² :d_∞(x,[0, l]× {0})≤a_ε} ∪Q_rε(0,0)∪Q_rε(l,0), Ibε :={x∈R² :d∞(x, Iaε)≤bε},

I_cε :={x∈R² :d∞(x,(I_aε∪I_bε))≤ε}, I_dε := Ω\(I_aε ∪I_bε ∪I_cε),

and define R_ε =I_aε \(Q_rε(0,0)∪Q_rε(l,0)).

∑ ∑

(0,0) (l,0)

∑ ∑

(0,0) (l,0)

Ib Ic

Ib Ic

B B

B B

Figure 3: Example of the neighborhoods of the segment [0, l]× {0}. On the left the case r_ε = ε on the right the case in which r_ε = a_ε > ε. The stripped region is R_ε and I_aε =R_ε∪(Q_rε(0,0)∪Q_rε(l,0)). Remark that supp(ρ_ε) =B(0, ε).

Costruction of σ_ε: We build σ_ε as a vector field supported on I_aε. In particular we add together three different constructions performed respectively on R_ε, Q_rε(0,0) and Q_rε(l,0). Letr =r_ε/ε and consider the problem







∆u=±θδ_x0 ∗ρ onQ_r(0,0),

∂u

∂ν = ±θ

H¹(Σ) on Σ^±={x∈R² :x₁ =±1, |x₂| ≤ θα 2 }.

∑⁺

(0,0)

Qr(0,0) B1(0,0)

∑^-

Letu⁺ be the solution relative to the problem in which every occurrence of ± is replaced by + and letu⁻ defined accordingly. Then set

σ_ε =



















∇u⁺(x/ε)

ε on Q_rε(0,0), θ

2a_ε ·e₁ on R_ε,

∇u⁻((x−(l,0))/ε)

ε on Q_rε(l,0).

(5.7)

By construction we have that ∇ ·σε =θ(δ_(0,0)−δ_(l,0))∗ρε and σε

* σ. Let us point out∗

as well that there exists a constant c(α, θ) such that c(α, θ) :=

Z

Qrε(l,0)

|σε|² dx= Z

Qrε(0,0)

|σε|² dx= Z

Qr(0,0)

∇u⁺(x)

2 dx= Z

Qr(0,0)

∇u⁻(x)

2 dx.

(5.8) Costruction of φ_ε: Most of the properties of φ_ε are a consequence of the inequalities obtained in Theorem 3.1 and the structure of σ_ε. On one hand we need φ_ε to attain the lowest value possible on I_aε in order to compensate the concentration of σ_ε in this set, on the other, as shown in inequality (3.6), we need to provide the optimal profile for the transition from this low value to 1. For this reasons we are led to consider the following ordinary differential equation associated with the optimal transition







w⁰_ε = 1

ε(1−w_ε), w_ε(0) =η.

(5.9)

Observe thatw_ε = 1−(1−η) exp ^−t_ε

is the explicit solution of equation (5.9) and set

φ_ε(x) :=











η if x∈I_aε,

w_ε(d∞(x, I_aε)) if x∈I_bε, d∞(x, I_bε)−ε+ 1 if x∈I_cε,

1 otherwise.

(5.10)

The choice of the behavior in the region Icε is given by the fact that following the optimal profile we will reach the value 1 only at +∞thus a linear correction on a small set ensures that this value is achieved with a small cost in energy.

Evaluation of Fε(σε, φε): We prove inequality (5.5) for the sequence we have produced.

Since the setsI_aε,I_bε, I_cε and I_dε are disjoint we can split the energy as follows

F_ε(σ_ε, φ_ε) =F_ε(σ_ε, φ_ε;I_aε) +F_ε(σ_ε, φ_ε;I_bε) +F_ε(σ_ε, φ_ε;I_cε) +F_ε(σ_ε, φ_ε;I_dε) (5.11) and evaluate each component individually. Since σ_ε is null and φ_ε is constant and equal to 1 in I_dε we have that F_ε(σ, φ_ε;I_dε) = 0. For the other components we strongly use the definitions in (5.7) and (5.10). Firstly we split again the energy on the setI_aε as following

F_ε(σ_ε, φ_ε;I_aε) =F_ε(σ_ε, φ_ε;R_ε) +F_ε(σ_ε, φ_ε;Q_rε(0,0)) +F_ε(σ_ε, φ_ε;Q_rε(l,0)).