VARIATIONAL APPROXIMATION OF SIZE-MASS ENERGIES FOR k-DIMENSIONAL CURRENTS

URL:http://www.emath.fr/cocv/

VARIATIONAL APPROXIMATION OF SIZE-MASS ENERGIES FOR k-DIMENSIONAL CURRENTS

^∗

Antonin Chambolle

¹

, Luca A. D. Ferrari

²

and Benoit Merlet

³

Abstract. In this paper we produce a Γ-convergence result for a class of energies F_ε,a^k modeled on the Ambrosio-Tortorelli functional. For the choicek= 1 we show thatF_ε,a¹ Γ-converges to a branched transportation energy whose cost per unit length is a functionf_aⁿ⁻¹ depending on a parametera >0 and on the codimension n−1. The limit costfa(m) is bounded from below by 1 +m so that the limit functional controls the mass and the length of the limit object. In the limita↓0 we recover the Steiner energy.

We then generalize the approach to any dimension and codimension. The limit objects are now k- currents with prescribed boundary, the limit functional controls both their masses and sizes. In the limit a↓0, we recover the Plateau energy defined onk-currents,k < n. The energiesFε,a^k then could be used for the numerical treatment of thek-Plateau problem.

R´esum´e. ...

1991 Mathematics Subject Classification. 49Q20; 49J45; 35A35.

...

1. Introduction

Let Ω⊂Rⁿ be a convex, bounded open set. We consider vector measuresσ∈ M(Ω,Rⁿ) of the form σ=m τH¹

x

^Σ,

where Σ is a 1-dimensional rectifiable set oriented by a Borel measurable tangent map τ : Σ → Sⁿ⁻¹ and m : Σ → R+ is a Borel measurable function representing the multiplicity. We write σ = (m, τ,Σ) for such

Keywords and phrases:Γ-convergence; Steiner problem; Plateau problem; Phase-field approximations

∗The authors have been supported by the ANR project Geometrya, Grant No. ANR-12-BS01-0014-01

1CNRS, CMAP, ´Ecole Polytechnique CNRS UMR 7641, Route de Saclay, F-91128 Palaiseau Cedex France.

e-mail: [email protected]

2CMAP, ´Ecole Polytechnique, CNRS UMR 7641, Route de Saclay, F-91128 Palaiseau Cedex France.

e-mail: [email protected]

3Laboratoire P. Painlev´e, CNRS UMR 8524, Universit´e Lille 1, F-59655 Villeneuve d’Ascq Cedex, France.

e-mail: [email protected]

©EDP Sciences, SMAI 1999

measures. Given a cost functionf ∈C(R+,R+) we introduce the functional

F(σ) :=





 Z

Σ

f(m) dH¹, ifσ= (m, τ,Σ), +∞, otherwise inM(Ω,R^d).

(1.1)

Next, given S = {x1,· · · , xn_P} ⊂ Ω a finite set of points and c1,· · ·, cn_P ∈ R such that Pn_P

j=1cj = 0, we consider the optimization problemF(σ) forσ∈ M(Ω,Rⁿ) satisfying

∇ ·σ =

nP

X

j=1

c_jδ_xj in D⁰(Rⁿ). (1.2)

The setting is similar to the one from Beckman [22] and Xia [25]. We model transport nets connecting a given set of sources {x_j ∈S : c_j >0} to a given set of wells {x_j ∈S : c_j <0} via vector valued measures. For numerical reasons, we wish to approximate the measureσ= (m, τ,Σ) by a diffuse object (a smooth vector field).

For this, we introduce below a family of corresponding “diffuse” functionals Fε,a that converge towards (1.1) in the sense of Γ-convergence [8, 9, 14]. This general idea has proved to be effective in a variety of contexts such as fracture theory, optimal partitions problems and image segmentation [3, 13, 18, 19]. More recently this tool has been used to approximate energies depending on one dimensional sets, for instance in [20] the authors take advantage of a functional similar to the one from Modica and Mortola defined on vector valued measures to approach the branched transportation problem [5]. With similar techniques approximations of the Steiner minimal tree problem ( [2, 17, 21]) have been proposed in [6, 7].

In the present paper we introduce a family of functionalsF_ε,ato extend to any ambient dimensionn≥2 the phase-fieldapproximation for a branched transportation energy introduced in [10] forn= 2. The approximating functionals are modeled on the ones from Ambrosio and Tortorelli [4] and allow to recover in the limit asε↓0 an energy of the form (1.1) for a particular choice of the cost functionf =fa that will be specified below. We also extend the construction to any dimension and co-dimension. Indeed, for 1≤k≤n−1 integer, we consider k-rectifiable currentsσ = (θ, e,Σ) where Σ is a countablyk-rectifiable set with approximate tangentk-plane defined by a simple unit multi-vectorξ(x) =ξ1(x)∧ · · · ∧ξk(x) andm: Σ→R+is a Borel measurable function (the multiplicity). The functional (1.1) extends tok-currentsσas follows,

F(σ) :=





 Z

Σ

f(m(x)) dH^k, ifσ= (m, ξ,Σ),

+∞, otherwise.

We will further show that for small values of a parameterawe can approach solutions to thek-Plateau problem.

Let us define the approximate functionals and describe our main results in the casek= 1. For our phase field approximations we relax the condition on the vector measureσreplacing it by a vector fieldσ_ε∈L²(Ω,Rⁿ). We then need to mollify condition (1.2). Letρ:Rⁿ→R+ be a classical radial mollifier such that suppρ⊂B₁(0) andR

B1(0)ρ= 1. Forε >0, we setρε=ε⁻ⁿρ(·/ε). We substitute for (1.2) the condition

∇ ·σ_ε =





nP

X

j=1

c_jδ_xj



∗ρ_ε=

nP

X

j=1

c_jρ_ε(· −x_j) in D⁰(Rⁿ). (1.3)

Remark 1.1. Notice that in (1.2) and (1.3) the equality holds in D⁰(Rⁿ) and not only in D⁰(Ω) so that there is no flux trough∂Ω.

We also consider the functionsu∈W^1,p(Ω,[η,1]) such thatu≡1 on∂Ω whereη=η(ε) satisfies η=a εⁿ

for some a ∈ R+. We denote by X_ε(Ω) the set of pairs (σ, u) such that uis as stated above and σ satisfies equation (1.3). This set is naturally embedded in M(Ω,Rⁿ)×L²(Ω). For (σ, u) ∈ M(Ω,Rⁿ)×L²(Ω) and p > n−1 we set

Fε,a(σ, u; Ω) :=





 Z

Ω

ε^p−n+1|∇u|^p+(1−u)²

εⁿ⁻¹ +u|σ|² ε

dx, if (σ, u)∈Xε(Ω),

+∞, in the other cases.

(1.4)

LetX be the subset ofM(Ω,Rⁿ)×L²(Ω) consisting of those couples (σ, u) such thatu≡1 andσ= (m, τ,Σ) satisfies the constraint (1.2). Given any sequenceε= (εi)_i∈Nof positive numbers such thatεi↓0, we show that the above family of functionals Γ-converges to

Fa(σ, u; Ω) =





 Z

Σ∩Ω

fa(m(x)) dH¹(x), if (σ, u)∈X and σ=m τH¹

x

^Σ,

+∞, otherwise.

(1.5)

The functionfa :R+→R+(introduced and studied in the appendix) is the minimum value of some optimization problem depending on a and on the codimension n−1 (we note f_a^d, with d = n−k in the general case 1≤k≤n−1). In particular we prove thatfa is lower semicontinuous, subadditive, increasing,fa(0) = 0 and that there exists somec >0 such that

1

c ≤ fa(m) 1 +√

a m ≤c form >0.

The Γ-convergence holds for the topology of the weak-∗convergence for the sequence of measures (σε) and for the strongL² convergence for the phase field (uε). For a sequence (σε, uε) we write (σε, uε)→(σ, u) ifσε

* σ∗

andkuε−uk_L2 →0. In the sequel we first establish that the sequence of functionals (Fε,a)ε is coercive with respect to this topology.

Theorem 1.2. Assume thata >0. For any sequence (σ_ε, u_ε)⊂ M(Ω,Rⁿ)×L²(Ω) withε↓0, such that F_ε,a(σ_ε, u_ε; Ω)≤F₀<+∞,

there existsσ∈ M(Ω,Rⁿ)such that, up to a subsequence, (σ_ε, u_ε)→(σ,1)∈X. Then we prove the Γ-liminf inequality

Theorem 1.3. Assume that a≥0. For any sequence(σε, uε)∈ M(Ω,Rⁿ)×L²(Ω) that converges to (σ, u)∈ M(Ω,Rⁿ)×L²(Ω)as ε↓0 it holds

lim inf

ε↓0 Fε,a(σε, uε; Ω)≥ Fa(σ, u; Ω).

We also establish the corresponding Γ-limsup inequality

Theorem 1.4. Assume that a≥0. For any (σ, u)∈ M(Ω,Rⁿ)×L²(Ω) there exists a sequence ((σ_ε, u_ε))⊂ M(Ω,Rⁿ)×L²(Ω)such that

(σε, uε)−→^ε↓0 (σ, u) in M(Ω,Rⁿ)×L²(Ω) and

lim sup

ε↓0

Fε,a(σε, uε; Ω)≤ Fa(σ, u; Ω).

As already stated, we only considered the case k = 1 in this introduction. Section 4 is devoted to the extension of Theorems 1.2, 1.3 and 1.4 in the case where the 1-currents (vector measures) are replaced with k-currents.

Notice that the coercivity of the family of functionals only holds in the casea > 0. However, as a↓ 0 we have the important phenomena:

f_a −→^a↓0 c1_(0,+∞) pointwise,

for somec >0. As a consequence (1.5) is an approximation of cH¹(Σ) fora >0 small and the minimization of (1.4) in X_ε(Ω) provides an approximation of the Steiner problem associated to the set of points S, for a suitable choice of the weights in (1.2). In the casek >1, we obtain a variational approximation of thek-Plateau problem.

Structure of the paper: In Section 2 we introduce some notation and recall some useful facts about vector measures and currents, we also anticipate the optimization problem defining the cost functionf_a^dand state some results which are proved in Appendix A. In Section 3 we establish Theorems 1.2, 1.3 and 1.4. In Section 4 we extend these results to the case 1≤k≤n−1. In Section 5 we discuss the limit a↓0.

2. Preliminaries and notation

The canonical orthonormal basis of Rⁿ is denoted by the vectors e1, . . . , en. Lⁿ denotes the Lebesgue measure inRⁿand given an integer valuekwe denote byωk the measure of the unit ball inR^k, i.e. L^k(B1(0)).

For a pointx∈Rⁿ we notex= (x1;x⁰)∈R×Rⁿ⁻¹. For any Borel-measurable setA⊂Rⁿ we denote by1A(x) the characteristic function of the setA

1A(x) :=

1 ifx∈A 0 otherwise.

Given a vector spaceY and its dualY⁰ forω∈Y andσ∈Y⁰ we writehω, σifor the dual pairing.

2.1. Measures and vector measures

We denote withM(Ω) the vector space of Radon measures in Ω and withM(Ω,Rⁿ) =M(Ω)ⁿ the vector space of vector valued measures. For a measureµ∈ M(Ω) we denote by |µ| its total variation, in the vector caseµ ∈ M(Ω,Rⁿ) we write µ= τ|µ| where τ is a |µ|-measurale map into Sⁿ⁻¹. We say that a measure is supported on a Borel setEif|µ|(Ω\E) = 0. For an integerk < nwe denote byH^k thek-dimensionalHausdorff measure as in [1]. Given a set E ∈ Ω, such that, H^k(E) is finite for some k the restriction H^k

x

^{E defines a}

Radon measure in the spaceM(Ω). A setE ∈Ω is said to be countablyk-rectifiable if up to aH^k negligible setN,E\N is contained in a countable union of C¹ k-dimensional manifolds.

2.2. Currents

We denote with D^k(Ω) the vector space of compactly supported smooth k-differential forms. For a k- differential formωits comass is defined as

kωk= sup{hω, ξi:ξis a unit, simplek-vector}.

LetDk(Ω) be the dual toD^k(Ω) i.e. the space ofk-currents with its weak-∗ topology. We denote with∂ the boundary operator that operates by duality as follows

h∂σ, ωi=hσ, dωi for all (k−1)-differential forms ω.

The mass of ak-currentM(σ) is the supremum of hσ, ωi among allk-differential forms with comass bounded by 1. For anyk-currentσsuch that bothσand∂σ are of finite mass we say thatσis a normal k-current and we writeσ∈N_k(Ω). On the spaceD_k(Ω) we can define theflat norm by

F(σ) = inf{M(R) +M(S) : σ=R+∂S whereS∈ Dk+1(Ω) and R∈ Dk(Ω)},

which metrizes the weak-∗topology on currents on compact subsets ofN_k(Ω). By the Radon-Nikodym theorem we can identify ak-currentσwith finite mass with the vector valued measureτ µ_σ whereµ_σ is a finite positive valued measure andτ is aµ_σ-measurable map in the set of unitaryk-vectors for the mass norm. In particular the action ofσonω can be written as

hσ, ωi= Z

Ω

hω, τidµσ.

For a finite massk-current the mass ofσcoincides with the total variation of the measureµσ. Ak-currentσis said to bek-rectifiable if we can associate to it a triplet (θ, τ,Σ) such that

hσ, ωi= Z

Σ

θhω, τidH^k

where Σ is a countablyk-rectifiable subset of Ω, τ at H^k a.e. point is a unit simple k-vector that spans the tangent plane to Σ andθis anL¹(Ω,H^k

x

Σ) function that can be assumed positive. We will denote withRk(Ω) the space of thesek-rectifiable currents. Among these we name out the subsetPk(Ω) of k-rectifiable currents for which Σ is a finite union of polyhedra andθ is constant on each of them, these will be calledpolyhedral chains. Finally theflat chains Fk(Ω) consist of the closure ofPk(Ω) in the weak-∗ topology. By the scheme of Federer [16, 4.1.24] it holds

Pk(Ω)⊂Nk(Ω)⊂Fk(Ω).

Remark 2.1 (1-Currents and Vector Measures). Since the vector spaces Λ1Rⁿ, Λ¹Rⁿ identify with Rⁿ, any vector measureσ∈ M(Ω,Rⁿ) with finite mass indentifies with a 1-current with finite mass and viceversa. The divergence operator acting on measures is defined by duality as the boundary operator for currents. In the following σ ∈ M(Ω,Rⁿ) is called a rectifiable vector measure if it is 1-rectifiable as 1-current. In the same fashion we define polyhedral 1-measures.

2.3. Functionals defined on flat chains

Forf :R7→R⁺an even function we define a functional Pk(Ω) −→ R+, P =X

j

(mj, τj,Σj)7−→ F(P) =X

j

f(mj)H^k(Σj),

on the space of polyhedral currents. Under the assumption thatf is lower semi-continuous and subadditive,F can be extended to a lower semi-continuous functional by relaxation

Fk(Ω) −→ R+,

P 7−→ F(P) = inf

lim inf

P_j→P F(P) : (Pj)j⊂Pk(Ω) andPj→P

as shown in [23, Section 6]. Furthermore, in [12] the authors show that iff(t)/t→ ∞ast→0, thenF(σ)<∞ if and only ifσis rectifiable and for any suchσthe functional takes the explicit form

F(σ) = Z

Σ

f(m(x)) dH^k(x) ifσ= (m, τ,Σ). (2.1)

To conclude this subsection let us recall a sufficient condition for a flat chain to be rectifiable, proved by White in [24, Corollary 6.1].

Theorem 2.2. Let σ ∈ Fk(Ω). If M(σ) +M(∂σ) < ∞ and if there exists a Borel set Σ ⊂ Ω with finite k-dimensional Hausdorff measure such thatσ=σ

x

^{Σthen σ∈Rk(Ω)i.e., σwrites as (m, τ,Σ).}

In the context of vector measures the theorem writes as

Theorem 2.3. Letσ∈ M(Ω,Rⁿ). If|σ|(Ω) +|∇ ·σ|(Ω)<∞,∇ ·σis at most a countable sum of Dirac masses and there exists a Borel setΣwithH¹(Σ)<∞andσ=σ

x

^Σthenσis a rectifiable vector measure in the sense expressed in Subsection 2.2.

2.4. Reduced problem results in dimension n − k

This subsection is devoted to introducing some notation and results corresponding to the case k = 1. In the sequel, these results are used to describe the energetical behaviour of the (n−k)-dimensional slices of the configuration (σ_ε, u_ε). We postpone the proofs to Appendix A.3, A.4 and A.5. We setd=n−k, p > d and considerεto be a sequence such thatε↓0. LetB_r(0)⊂R^d be the ball of radiusrcentered in the origin. We consider the functional

Eε,a(ϑ, u;Br) :=

Z

B_r

ε^p−d|∇u|^p+(1−u)²

ε^d +u|ϑ|² ε

dx

where u∈ W^1,p(B_r) is constrained to satisfy the lower bound u ≥a ε^d+1 =: η and ϑ∈ L²(B_r) is such that supp(ϑ)⊂B_r˜with 0<r < r,˜ kϑk1=m. This leads to define the set

Yε,a(m, r,r) =˜

(ϑ, u)∈L²(Br)×W^1,p(Br,[η,1]) : kϑk1=mand supp(ϑ)⊂B˜r , and the optimization problem

f_ε,a^d (m, r,r) =˜ inf

Y_ε,a(m,r,˜r)Eε,a(ϑ, u;Br). (2.2) Letf_a^d: [0,+∞)−→R+be defined as

f_a^d(m) =









 minˆr>0

a m²

ωdrˆ^d + ωd ˆr^d+ (d−1)ωdq_∞^d (0,r)ˆ

, form >0,

0, form= 0,

(2.3)

with

q^d_∞(ξ,r) := infˆ

Z +∞

ˆ r

t^d−1

|v⁰|^p+ (1−v)²

dt : v(ˆr) =ξ and lim

t→+∞v(t) = 1

, (2.4)

for ˆr > 0,ξ ≥ 0. For a graph of the profilev realizing the infimum in the latter see Figure 4. We have the following results

Proposition 2.4. For anyr >r >˜ 0, it holds lim inf

ε↓0 f_ε,a^d (m, r,r)˜ ≥f_a^d(m). (2.5)

There exists a uniform constantκ:=κ(d, p)such that

f_a^d(m)≥κ for everym >0. (2.6)

Proposition 2.5. For fixed m > 0 let r_∗ be the minimizing radius in the definition of f_a^d(m) (2.3). For any δ > 0 and ε small enough there exist a function ϑ_ε = c1_Br∗ε with c > 0 such that R

Brϑ_ε = m and a nondecreasing radial functionuε:Br7→[η,1]such thatuε(0) =η,uε= 1 on∂Br and

E_ε,a(ϑ_ε, u_ε;B_r)≤f_a^d(m) +δ. (2.7) Proposition 2.6. The functionf_a^d is continuous in (0,+∞), increasing, sub-additive and f_a^d(0) = 0.

3. The 1 -dimensional problem

3.1. Compactness

We prove the compactness Theorem 1.2 for the family of functionals (Fε,a)ε. Let us consider a family of functions (σε, uε)_ε↓0, such that (σε, uε)∈Xε(Ω) and

Fε,a(σε, uε; Ω) ≤ F0. As a first step we prove:

Lemma 3.1. Assume a >0. There existsC≥0, depending only on Ω,F₀ andasuch that Z

Ω

|σε| ≤ C, ∀ε. (3.1)

As a consequence there exist a positive Radon measure µ ∈ (Rⁿ,R+) supported in Ω and a vectorial Radon measureσ∈ M(Ω,Rⁿ)with ∇ ·σ=Pa_jδ_xj and|σ| ≤µsuch that up to a subsequence

uε→1 inL²(Ω), |σε| * µ^∗ inM(Rⁿ), σε

* σ∗ in M(Rⁿ,Rⁿ).

Proof. We divide the proof into three steps.

Step 1. We start by proving the uniform bound (3.1). Letλ∈(0,1] and let Ωλ := {x∈Ω : uε> λ}.

Beingσ_εsquare integrable we identify the measureσ_εwith its density with respect toLⁿ. Therefore splitting the total variation ofσ_ε, we write

|σε|(Ω) = Z

Ω

|σε|dx= Z

Ωλ

|σε|dx+ Z

Ω\Ωλ

|σε|dx.

We estimate each term separately. By Cauchy-Schwarz inequality we have Z

Ω_λ

|σε| ≤ Z

Ω_λ

uε|σε|² ε

^1/2Z

Ω_λ

ε uε

1/2

.

Sinceλ < u_ε≤1 on Ω_λ andR

Ωλ(u_ε|σε|²)/(ε) dxbeing bounded byFε,a(σ_ε, u_ε) from the previous we get Z

Ωλ

|σε| ≤ Z

Ωλ

uε|σε|² ε

^1/2r

|Ω|ε

λ ≤

r|Ω|ε F0

λ .

Next, in Ω\Ω_λ, by Young inequality, we have 2

Z

Ω\Ωλ

|σε| ≤ Z

Ω\Ωλ

uε|σε|²

ε +

Z

Ω\Ωλ

ε u_ε. Usingu_ε≥η(ε),η/εⁿ=aand (1−λ)²≤(1−u_ε)²in Ω\Ω_λ, we obtain

Z

Ω\Ωλ

|σε| ≤ 1 2

Z

Ω

uε|σε|²

ε + εⁿ

2η(1−λ)² Z

Ω

(1−uε)² εⁿ⁻¹ ≤F0

2 + F0

2a(1−λ)². Hence

|σ_ε|(Ω)≤ F0

2 + F0

2a(1−λ)² +

r|Ω|ε F0

λ .

Asa >0, this yields (3.1).

Step 2. We easily see fromR

Ω(1−u_ε)² ≤ F₀εⁿ⁻¹that u_ε→1 inL²(Ω) asε↓0.

Step 3. The existence of the Radon measuresµ and σ such that, up to extraction, |σε| * µ^∗ and σε

* σ∗

follows from (3.1). The properties on the support ofµ, on the divergence ofσ and the fact that|σ| ≤µfollow

from the respective properties ofσε.

We have just showed that the limit σ of a family (σε, uε)ε equibounded in energy is bounded in mass. In what follows, we assumea≥0 and thatσεis bounded in mass. We show that the limitingσis rectifiable.

Proposition 3.2. Assumea≥0 and that the conclusions of Lemma 3.1 hold true. There exists a Borel subset Σwith finite length and a Borel measurable function τ : Σ→Sⁿ⁻¹ such thatσ=τ|σ|

x

Σ. Moreover, we have the following estimate,

H¹(Σ) ≤ C_∗F0, where the constantC_∗≥0only depends on dandp.

This proposition together with Lemma 3.1 and Theorem 2.3 leads to

Proposition 3.3. σis a1-rectifiable vector measure and in particularΣis a countablyH¹-rectifiable set.

The latter ensures that the limit couple (σ,1) belongs toX and concludes the proof of Theorem 1.2. We now establish Proposition 3.2

Sketch of the proof: We first define Σ. Then we show in Lemma 3.5 that forx∈Σ, we have

lim inf_ε↓0Fε,a(σε, uε;B(x, rj))≥κrj for a sequence of radiirj↓0 andκ >0. The proof of the lemma is based on slicing and on the results of Appendix A. The proposition then follows from an application of the Besicovitch covering theorem.

First we introduce the Borel set Σ :=e

x∈Ω : ∀r >0, |σ|(Br(x))>0 and∃τ=τ(x)∈Sⁿ⁻¹such thatτ= lim

r↓0

σ(B_r(x))

|σ|(Br(x))

. We observe that by Besicovitch derivation theorem,

σ=τ|σ|

x

^Σ.e

Next we fixθ∈(0,1/4ⁿ) and define Γ :=

x∈Σ :e ∃r0>0 such that, |σ|(B_r/4(x))

|σ|(Br(x)) ≤θ for every r∈(0, r0]

. We show that this set is|σ|-negligible.

Lemma 3.4. We have|σ|(Γ) = 0.

Proof. Let x ∈ Γ. Applying the inequality |σ|(B_r/4(x)) ≤ θ|σ|(Br(x)) with r = rk = 4^−kr0, k ≥ 0, we get

|σ|(Br_k) ≤ θ^k|σ|(Br0). Hence there existsC≥0 such that

|σ|(Br(x)) ≤ Cr(ln 1/θ)/(ln 4)

.

Lettingλ= (ln¹_θ)/(ln 4), we have by assumptionλ > n. Therefore, for every ξ >0 there exists rξ =rξ(x)∈ (0,1) such that

|σ|(Br_ξ(x)) ≤ ξ|Br_ξ(x)|.

Now, forR >0, we cover Γ∩BRwith balls of the formB_rξ(x)(x). Using Besicovitch covering theorem, we have Γ∩BR ⊂ ∪^N(n)_j=1 Bj

where N(n) only depends on n and each B_j is a (finite or countable) disjoint union of balls of the form B_rξ(xk)(xk). Then we get

|σ|(Γ∩B_R) ≤

N(n)

X

j=1

|σ|(B_j) ≤ N(n)ξ|B_j| ≤ N(n)|B_R+1|ξ.

Sendingξto 0 and thenR to∞, we obtain|σ|(Γ) = 0.

Set Σ := Σe\Γ, from Lemma 3.4, we haveσ=τ|σ|

x

^Σ.Recall that S ={x1,· · ·, xn_P}.

Lemma 3.5. For everyx∈Σ\S, there exists a sequence(r_j) = (r_j(x))⊂(0,1)with r_j↓0such that lim inf

ε↓0 Fε,a(σε, uε;B(x, rj)) ≥ √ 2κ rj, whereκis the constant of Proposition 2.4.

Proof. Let x ∈ Σ\S. Without loss of generality, we assume x = 0 and τ(x) = e1. Let ξ > 0 be a small parameter to be fixed later. From the definition of Σ, there exists a sequence (rj) = (rj(x))⊂(0, d(x,S)) such that for everyj≥0,

σ(Br_j)·e1 ≥ (1−ξ)|σ|(Br_j) and |σ|(B_rj/4) ≥ θ|σ|(Br_j). (3.2) Let us fixj ≥0 and set, to simplify the notation, r=r_j andr_∗ =r/√

2. Recall the notation x= (x₁, x⁰)∈ R×Rⁿ⁻¹ and define the cylinder

Cr∗:={x:|x1| ≤r_∗ and |x⁰| ≤r_∗}

so thatCr∗⊂BrandBr/4⊂Cr_∗/2, as shown in figure 1. Letχ∈C_c^∞(Rⁿ⁻¹,[0,1]) be a radial cut-off function such thatχ(x⁰) = 1 if|x⁰| ≤ ¹₂ andχ(x⁰) = 0 for|x⁰| ≥ ³₄. Then, we noteχr_∗(x⁰) =χ(x⁰/r_∗) and fors∈[−r, r], we set

∀s∈[−r, r], gε(s) := e1· Z

B_r∗⁰

σε(s, x⁰)χr∗(x⁰) dx⁰.

Sinceσεis divergence free,e1·σε(·, s) has a meaning on the hyperplane{x1=s}in the sense of trace, moreover, gε is continuous. Now, let us fix ˆr ∈ [(1−ξ)r∗, r∗] such that µ({−ˆr,r} ׈ B⁰_r) = 0 (which holds true for a.e.

ˆ

r∈[(1−ξ)r∗, r∗]) and let us define the mean value,

¯ g_ε := 1

2ˆr Z rˆ

−ˆr

g_ε(s) ds.

e1

ˆ r

Br/4 Cr∗

Br

O

Figure 1. Illustration of the sections ofBr,B_r/4 andCr_∗. In grayscale we represent the level sets of the functionχr_∗(x⁰)1_[−ˆr,ˆr].

Fromσε

* σ,∗ |σε| * µ, we have^∗

lim

ε↓0¯g_ε = 1 2ˆr

Z

(−ˆr,ˆr)×B_r∗⁰

χ_r∗(x⁰) dσ(s, x⁰)

!

·e₁ =: m.

From (3.2), we see thatm >0 for ξsmall enough. Indeed, we have (1−ξ)|σ|(Br) ≤ +2ˆrm+σ(Br)·e1 =

Z

Br

1−χr∗(x⁰)1_[−ˆr,ˆr]

dσ(s, x⁰)·e1

≤ 2ˆrm+ Z

B_r

1−χ_r∗(x⁰)1_[−ˆr,ˆr]

d|σ|(s, x⁰)

≤ 2ˆrm+|σ|(Br)− Z

Br

χr∗(x⁰)1_[−ˆr,ˆr] d|σ|(s, x⁰).

Since by constructionχr∗(x⁰)1_[−ˆr,ˆr]≥1B_r/4, using the second inequality of (3.2), we have m ≥ 1

2ˆr(θ−ξ)|σ|(Br) > 0,

forξ small enough. Similarly, denoting Π : Rⁿ →Rⁿ⁻¹, (t, x⁰)7→x⁰ the orthogonal projection onto the last (n−1) coordinates, we deduce again from (3.2) that

|Πσ|(Cr∗)≤

√ξm

θ−ξ2ˆr. (3.3)

Now, forε small enough, we have∇ ·σ_ε = 0 in C_r∗. Using this, we have for almost everys, t∈ [−ˆr,r], withˆ s < t,

gε(t)−gε(s) = Z t

s

"

Z

B_r∗⁰

σε(x⁰, h)· ∇⁰χr∗(x⁰) dx⁰

# dh.

Integrating insover (−ˆr,r), we get for almost everyˆ t∈[−r, r], gε(t)−¯gε = 1

2ˆr Z

(−ˆr,ˆr)×B_r∗⁰

φt(x⁰, h)·σε(x⁰, h) dx⁰ dh

with

φ_t(h, x⁰) =

((h+ ˆr)∇⁰χ_r∗(x⁰) ifh < t, (h−r)ˆ ∇⁰χ_r∗(x⁰) ifh > t.

We deduce the following convergence

gε(t)−m−→^ε↓0 1 2ˆr

Z

(−ˆr,ˆr)×B_r∗⁰

φt(h, x⁰)· dσ(h, x⁰).

in theL¹(−ˆr,ˆr) topology. Using (3.3), we see that the above right hand side is bounded by c

√ξ

θ−ξm. Taking into account (3.3) and the continuity ofgε, we conclude that

lim inf

ε↓0 g_ε(t)≥

1−c

√ξ θ−ξ

m for t∈[−ˆr,r].ˆ Next, by decomposing the integral we have

Fε,a(σε, uε;Br) ≥ Z ˆr

−ˆr

Z

B_r⁰

∗

ε^p−n+1|∇uε|^p+(1−uε)²

εⁿ⁻¹ +uε|σε|² ε

dx⁰ dt

≥ Z ˆr

−ˆr

Z

B_r⁰

∗

ε^p−n+1|∇uε|^p+(1−uε)²

εⁿ⁻¹ +uε|χr∗(x⁰)σε|² ε

dx⁰ dt.

Let us set

ϑ^t_ε(x⁰) :=|χr∗(x⁰)σ_ε(t, x⁰)|.

By constructionϑ^t_εhas the properties:

• ϑ^t_ε∈L¹(B_r⁰_∗),

• lim inf_ε↓0R

B_r∗⁰ ϑ^t_ε(x⁰)dx⁰ ≥ lim inf_ε↓0g_ε(t) ≥ 1−c

√ξ θ−ξ

m= ˜m >0,

• supp(ϑ^t_ε)⊂B_r⁰_˜with ˜r:= ³₄r_∗< r_∗.

By definition of the minimization problem introduced in Subsection 2.4, we have Fε,a(σε, uε;Br) ≥

Z rˆ

−ˆr

inf

(ϑ,u)∈Yε,a( ˜m,r,˜r)

Eε,a(ϑ, u;Br)

dt= Z rˆ

−ˆr

fε,a( ˜m, r,r) dt.˜ Taking the infimum limit, by Fatou’s lemma and equation (2.6) of Proposition 2.4 we get

lim inf

ε↓0 Fε,a(σε, uε;Br)≥ Z ˆr

−ˆr

lim inf

ε↓0 fε,a( ˜m, r,r) dt˜ ≥ 2 ˆr κ.

The latter holds for almost every ˆr∈[(1−ξ)r_∗, r_∗] and eventually, since ther_∗=r/√

2, we conclude lim inf

ε↓0 F_ε,a(σ_ε, u_ε;B_r)≥ √ 2κ r.

The proof of Proposition 3.2 is then obtained via the Besicovitch covering theorem [15].

3.2. Γ-liminf inequality

In this subsection we prove the Γ−lim inf inequality stated in Theorem 1.3.

Proof of Theorem 1.3. With no loss of generality we assume that lim inf_ε↓0Fε,a(σε, uε) <+∞ otherwise the inequality is trivial. For a Borel setA⊂Ω, we define

H(A) := lim inf

ε↓0 Fε,a(σε, uε;A),

so that H is a subadditive set function. By assumption, the limit measure σ is 1-rectifiable; we write σ = m τH¹

x

Σ. Furthermore we can assumeσto be compactly supported in Ω. Consider a convex open set Ω0such that supp(∇ ·σ) =S ⊂⊂Ω₀⊂⊂Ω and leth:= [0,1]×Rⁿ→Rⁿ be a smooth homotopy of the indentity map onRⁿ onto a contraction of Ω into Ω0 such thath(t,·) restricted to Ω0 is the identity map, for anyt ∈[0,1].

Letσt = h(t,·)]σ, indeed lim inf_t↓0F(σt,1) ≥ F(σ,1) as σt

* σ. Further∗ ∇ ·σt =∇ ·σ since h(t,·) is the identity onS. Now we claim that

lim inf

r↓0

H

B(x, r)

2r ≥ fa(m(x)) forH¹-almost everyx∈Σ. (3.4) Let us fixλ≥1 and let us note f_a,λ(t) := min(f_a(t), λ). We then introduce the Radon measure

H_λ⁰(A) :=

Z

Σ∩A

f_a,λ(m) dH¹.

Now, letδ∈(0,1). Assuming that (3.4) holds true, there exists Σ⁰⊂Σ withH¹(Σ\Σ₀) = 0 such that for every x∈Σ₀, there existsr₀(x)>0 with

(1 +δ)H

B(x, r)

≥ 2rf_a,λ(m(x)) for everyr∈(0, r₀(x)).

By the Besicovitch differentiation Theorem, there exists Σ₁⊂Σ withH¹(Σ\Σ1) = 0 such that for everyx∈Σ₁, there existsr₁(x)>0 with

(1 +δ)2rfa(m(x)) ≥ H_λ⁰

B(x, r)

for everyr∈(0, r1(x)).

We consider the famillyBof closed ballsB(x, r) withx∈Σ0∩Σ1 and 0< r <min(r0(x), r1(x)) and we apply the Vitali-Besicovitch covering theorem [1, Theorem 2.19.] to the familyBand to the Radon measureH_λ⁰. We obtain a disjoint family of closed ballsB⁰⊂ Bsuch that

H_λ⁰(Ω) =H_λ⁰(Σ) = X

B(x,r)∈B⁰

H_λ⁰

B(x, r)

≤ (1 +δ)² X

B(x,r)∈B⁰

H

B(x, r)

≤ (1 +δ)²H(Ω).

Sendingλto infinity and thenδto 0, we get the lower boundH(Ω)≥R

Σf_a(m)dH¹which proves the theorem.

Let us now establish the claim (3.4). Sinceσis a rectifiable measure, we have for H¹-almost everyx∈Σ, 1

2r Z

ϕ(x+ry) d|σ|(y) −→^r↓0 m(x) Z

R

ϕ(tτ(x)) dt for every ϕ∈Cc(Rⁿ), (3.5)

and 1

2r Z

B(x,r)∩Σ

|τ(y)−τ(x)| d|σ|(y) −→^r↓0 0. (3.6)

Letx∈Σ\ Sbe such a point. Without loss of generality, we assumex= 0,τ(0) =e₁andm:=m(0)>0. Let δ∈(0,1). Our goal is to establish a precise lower bound forF_ε,a(σ_ε, u_ε;C) whereCis a cylinder of the form

C_r^δ := {x∈Rⁿ : |x₁|< δr,|x⁰|< r}.

For this we proceed as in the proof of Lemma 3.5, here, the rectifiability ofσ simplifies the argument. Let χ^δ ∈C_c^∞(Rⁿ⁻¹,[0,1]) be a radial cut-off function withχ^δ(x⁰) = 1 if|x⁰| ≤δ/2,χ^δ(x⁰) = 0 if|x⁰| ≥δ. Forε >0 andr∈(0, d(0, ∂Ω)), we define fors∈(−r, r),

g^δ,r_ε (s) := e₁· Z

Rⁿ⁻¹

σ_ε(s, x⁰)χ^δ(x⁰/r) dx⁰. We also introduce the mean value

g^δ,rε := 1 2r

Z r

−r

g_ε^δ,r(s) ds.

From (3.5), we have forr >0 small enough, g₀^δ,r := 1

2r Z r

−r

e1· Z

Rⁿ⁻¹

σε(s, x⁰)χ^δ(x⁰/r) dx ds ≥ (1−δ)m.

For suchr >0, we deduce fromσε

* σ∗ that forε >0 small enough gε^δ,r := 1

2r Z r

−r

g_ε^δ,r(s) ds ≥ (1−2δ)m. (3.7)

We study the variation ofg_ε^δ,r(s). Using∇ ·σ_ε= 0 inC_r^δ, we compute as in the proof of Lemma 3.5, g_ε^δ,r(t)−g^δ,rε = 1

2r Z

(−r,r)×Bδr

φt(x⁰, h)·σε(x⁰, h) dx⁰ dh with

φ_t(h, x⁰) =

((h+ ˆr)∇⁰χ^δ(x⁰/r) ifh < t, (h−r)ˆ ∇⁰χ^δ(x⁰/r) ifh > t.

Using again the convergenceσε

* σ, we deduce∗

g^δ,r_ε (t)−gε^δ,r

−→ε↓0 1 2r

Z

(−r,r)×B_δr

φ_t(x⁰, h)· dσ(x⁰, h),

inL¹(−r, r). Now, sincee₁· ∇⁰χ^δ≡0, we deduce from (3.6) that the right hand side goes to 0 asr↓0. Hence, forr >0 small enough,

1 2r

Z

(−r,r)×Bδr

φt(x⁰, h)·σ(x⁰, h) dx⁰ dh

≤ δm.

Using (3.7), we conclude that forr >0 small enough and then forε >0 small enough, we have g^δ,r_ε (t) ≥ (1−3δ)m, for a.e. t∈(−r, r).

By definition of the codimension-0 problem, we conclude that

F_ε,a(σ_ε, u_ε;C_r^δ) ≥ 2rf_ε,aⁿ⁻¹((1−3δ)m).

Sendingε↓0 and recalling that we omit the superscript when it isn−1, we obtain H(C_r^δ) ≥ 2rfa((1−3δ)m).

We notice thatH(B^√_1+δ2r)≥H(C_r^δ). Dividing by 2√

1 +δ²r and taking the liminf asr↓0, we get lim inf

r↓0

H(B^√_1+δ2r) 2√

1 +δ²r ≥ fa((1−3δ)m)

√1 +δ² .

Sendingδto 0, we get (3.4) by lower semi-continuity off_a.

3.3. Γ-limsup inequality

Proof of Theorem 1.4.

Let us supposeF(σ, u; Ω)<+∞, so that in particularu≡1. From Xia [26], we can assumeσto be supported on a finite union of compact segments and to have constant multiplicity on each of them, namely polyhedral vector measures are dense in energy. We first construct a recovery sequence for a measureσconcentrated on a segment with constant multiplicity. Then we show how to deal with the case of a polyhedral vector measures.

Step 1. (σ concentrated on a segment.) Assume that σis supported on the segmentI = [0, L]× {0} and writes asm·e1H¹

x

^{I. Considerm}constant so that∇ ·σ=m(δ_(0,0)−δ_(L,0)) and

F(σ,1; Ω) =f_a(m)H¹(I) =L f_a(m).

Forδ >0 fixed, we consider the profiles

uε(t) :=











η, for 0≤t≤r_∗ε, vδ

t ε

, forr_∗ε≤t≤r,

1 forr≤t,

and ϑε= m χB⁰_r∗ε(x⁰) ω_n−1(εr_∗)ⁿ⁻¹

withr_∗andvδ, defined in Proposition 2.5 withd=n−1. Assumer_∗≥1 and letd(x, I) be the distance function from the segmentI and introduce the sets

Ir∗ε:={x∈Ω : d(x, I)≤r∗ε}, and Ir:={x∈Ω : d(x, I)≤r}.

Setu_ε(x) =u_ε(d(x, I)) andσ¹_ε=mH¹

x

^{I∗ρε, whereρε}is the mollifier of equation (1.3). We first construct the vector measures

σ¹_ε=σ¹_εe1 and σ_ε²(x1, x⁰) =ϑε(|x⁰|)e1. Alternatively,σ_ε²=σ∗ρ˜_εfor the choice ˜ρ_ε(x₁, x⁰) =χ_B⁰

r∗ε(x⁰)/ ω_n−1(εr_∗)ⁿ⁻¹. Let us highlight some properties ofσ_ε¹andσ²_ε. Both vector measures are radial inx⁰, with an abuse of notation we denoteσ¹_ε(x1, s) =σ¹_ε(x1,|x⁰|).

Since, bothσ¹_εandσ²_εare obtained trough convolution it holds supp(σ_ε¹)∪supp(σ_ε²)⊂Ir∗εand they are oriented by the vectore1 therefore|σ_ε¹|=σ¹_ε and|σ²_ε|=ϑε. Furthermore for anyx1, it holds

Z

{x1}×B⁰_r∗ε

σ¹_ε(x1, x⁰)−ϑε(x⁰)

dx⁰= 0 (3.8)

We construct σ_ε by interpolating between σ¹_ε and σ_ε². To this aim consider a cutoff function ζ_ε : R → R+

satisfying

ζε(t) = 1 fort≤r∗εor t≥L−r∗ε,

ζε(t) = 0 for 2r_∗ε≤t≤L−2r_∗ε, and |ζ_ε⁰| ≤ 1 r_∗ε. and set







σ_ε³·e₁= 0,

σ_ε³·ei(x1, x⁰) =−ζ_ε⁰(x1) xi

|x⁰|ⁿ⁻¹ Z |x⁰|

0

sⁿ⁻²

σ¹_ε(x1, s)−ϑε(s)

ds, fori= 2, . . . , n.

The integral corresponds to the difference of the fluxes of σ_ε¹ and σ²_ε through the (n−1)-dimensional disk {x₁} ×B⁰. Forσ_ε³ we have the following

∇ ·σ_ε³=−ζ_ε⁰(x1)

n

X

i=2

"

1

|x⁰|ⁿ⁻¹ −(n−1)x²_i

|x⁰|ⁿ⁺¹

Z |x⁰| 0

sⁿ⁻²

σ¹_ε(x1, s)−ϑε(s) ds + x²_i

|x⁰|²

σ¹_ε(x₁,|x⁰|)−ϑ_ε(|x⁰|)

=−ζ_ε⁰(x₁)

σ¹_ε(x₁,|x⁰|)−ϑ_ε(|x⁰|)

. (3.9) Let

σ_ε=ζ_εσ_ε¹+ (1−ζ_ε)σ_ε²+σ_ε³. In force of equation (3.9) and from the construction ofσ_ε¹,σ_ε² andζε we have

∇ ·σε=∇ ·(ζεσ_ε¹) +∇ ·(1−ζε)σ²_ε+∇ ·σ_ε³

=ζε∇ ·σ¹_ε+ζ_ε⁰(σ¹_ε−ϑε) +∇ ·σ³_ε

=ζε∇ ·σ¹_ε=∇ ·(σ∗ρε).

In addition for any (x1, x⁰) such that|x⁰| ≥r∗εfrom (3.8) we derive σ_ε³·ei(x1, x⁰) =−ζ_ε⁰(x1) xi

|x⁰|ⁿ⁻² Z |x⁰|

0

sⁿ⁻¹

σ¹_ε(x1, s)−ϑε(s) ds= 0 which justifies supp(σ_ε)⊂I_r∗ε. Let us now prove

lim sup

ε↓0

Fε,a(σε, uε; Ω)≤L fa(m) +Cδ.

We split Ω as the union of Ω\Ir, Cr,ε := Ir∩[2ε, L−2 ε]×Rⁿ⁻¹ and Dε and D_ε⁰, as show in figure 2, whereD_ε={x1≤2r_∗ε} ∩I_r∗εandD⁰_ε={x1≥L−2r_∗ε} ∩I_r∗ε. On Ω\I_rwe notice thatσ_ε= 0 andu_ε= 1 therefore

Fε,a(σε, uε; Ω\Ir) = 0.

Observe that|Dε|=|D⁰_ε|=Cεⁿ, then we have the upper bound Z

Dε

|σε|²dx≤2 m²r_∗² εⁿ⁻²

Z

B1

ρ²dx+C

.

Taking into consideration this estimate we obtain

Fε,a(σε, uε;Dε) =Fε,a(σε, uε;D_ε⁰)≤ (1−η)²

εⁿ⁻¹ Lⁿ(Dε) + 2m²r²_∗ η

εⁿ⁻². (3.10)

O εr∗

2εr∗ L-2εr∗

L-εr∗ Ir∗ε

I_r

Dε D_ε⁰

Ω

L

Figure 2. Illustration of the intervalI and both its r and (r_∗ε)-enlargement forr_∗ ≥1. In grayscale we plot the levels of the function ζ_ε, whilst the striped region corresponds to the cylinderC_r,ε.

Finally on Cr,ε both σε and uε are independent of x1 and are radial in x⁰ then by Fubini’s theorem and Proposition 2.5 we get

Fε,a(σε, uε;Cr,ε) =

Z L−2εr_∗ 2εr∗

Z

B⁰_r

Eε,a(ϑε, uε)≤ L(fa(m) +C δ).

Adding all together gives the desired estimate. It remains to discuss the caser_∗<1. From the point of view of the construction ofσεwe need to replace the functionsζε with

ζ˜ε(t) = 1 fort≤εort≥L−ε,

ζ˜ε(t) = 0 for 2ε≤t≤L−2ε, and

ζ˜_ε⁰ ≤ 1

ε.

This choice ensures thatσ_ε has all the properties previously obtained with r_∗ε replaced byε accordingly.

Define

w_ε(t) :=







η, fort≤√

3ε 1−η

r−√

3(t−√

3) +η, for√

3ε≤t≤r.

and set

u_ε= min{uε(d(x, I)), w_ε(|x|), wε(|x−(L; 0)|)}.

with these choices foru_εandσ_εthe estimates follow analogously with small differences in the constants.

Step 2. (Case of a genericσin polyhedral form.) Indeed, in force of the results quoted in Subsection 2.3 it is sufficient to show equation (1.4) for a polyhedral vector measure. Following the same notation introduced therein let

σ=

N

X

j=1

m_jH¹

x

^{Σj τj.}

With no loss of generality we can assume that the segments Σ_j intersect at most at their extremities. We consider measuresσ satisfying constraint (1.2) so that if a point P belongs to Σ_j1, . . . ,Σ_jP it must satisfy of Kirchhoff law,

jP

X

j1

zjmj=

c_i, ifP∈S.

0, otherwise. (3.11)

O ε

2ε L-2ε

L-ε

L O

ε

2ε L-2ε

L-ε L B^√_3ε(0; 0)

B^√_3ε(L; 0) Ir

Ir∗ε

Dε D⁰_ε

Ω Ir

Ir∗ε

Dε D⁰_ε

Figure 3. On the left the striped region corresponds to supp(σε), remark that the balls of radius√

3εcentered respectively in (0; 0) and (L; 0) contain the modifications we have performed to satisfy the constraint. On the right we illustrate the level-lines of the cutoff function ˜ζε in grayscale.

wherez_j, is +1 ifP is the ending point of the segment Σ_j with respect to its orientation, and−1 if it is the starting point. Letσ_ε^j andu^j_εbe the sequences constructed above for each segmentIk and define

σ_ε=

N

X

j=1

σ^j_ε and u_ε= min

j

u^j_ε .

LetPj andQj be respectively the initial and final point of the segment Σj and recall that, by the construction made above, for eachj

∇ ·σ_ε^j=m_j δ_Pj−δ_Qj

∗ρ_ε then by linearity of the divergence operator, it holds

∇ ·σε=

N

X

j=1

∇ ·σ_ε^k=

N

X

j=1

mj δP_j−δQ_j

∗ρε

and the latter satisfies constraint (1.3) in force of equation (3.11). To conclude let us prove that

lim sup

ε↓0

Fε,a(σε, uε; Ω)≤

N

X

j=1

fa(mj)H¹(Σj). (3.12)

Indeed the following inequality holds true

F_ε,a(σ_ε, u_ε; Ω)≤

N

X

j=1

F_ε,a(σ_ε, u^j_ε; Ω).

Suppose

supp(σ_ε^j1)∩supp(σ_ε^j2)∩ · · · ∩suppσ_ε^jP 6=∅

for some j₁, . . . , j_P and all ε. Let r^j_∗¹, . . . , r_∗^jP be the radii introduced above for each of these measures, let r_∗= max{r^j_∗¹, . . . , r_∗^jP,1} , setm= max{mj₁, . . . , mj_P}and considerDj₁, . . . , Dj_P as defined previously. Since

jP

X

k=1

σ_ε^k

2

≤C

jP

X

k=1

σ_ε^k

2