, Luca A.D. Ferrari

https://doi.org/10.1051/cocv/2018027 www.esaim-cocv.org

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 energiesFε,a^k modeled on the Ambrosio-Tortorelli functional. For the choicek= 1 we show thatFε,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 codimensionn−1. The limit costfa(m) is bounded from below by 1 +mso 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.

Mathematics Subject Classification.49Q20, 49J45, 35A35.

Received October 21, 2017. Accepted April 17, 2018.

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, (1.1)}

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

F(σ) :=





 Z

Σ

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

(1.2)

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

Keywords and phrases:Γ-convergence, Steiner problem, plateau problem, phase-field approximations.

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

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

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

∗∗Corresponding author:[email protected]

Article published by EDP Sciences ©EDP Sciences, SMAI 2019

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 problem F(σ) forσ∈ M(Ω,Rⁿ) satisfying

∇ ·σ=

n_P

X

j=1

cjδx_j inD⁰(Rⁿ). (1.3)

The setting is similar to the one from Beckman [22] and Xia [25]. We model transport nets connecting a given set of sources {xj ∈S : cj >0} to a given set of wells {xj ∈S : cj <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 (1.2) 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 [1,17,21] 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-field approximation for a branched transportation energy introduced in [10] forn= 2. The approximating functionals are modeled on the ones from Ambrosio and Tortorelli [2] and allow to recover in the limit asε↓0 an energy of the form (1.2) 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 currents σ= (θ, e,Σ) where Σ is a countably k-rectifiable set with approximate tangent k-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 (1.2) 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 fieldσε∈L²(Ω,Rⁿ). We then need to mollify condition (1.3). Let ρ: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 (1.3) the condition

∇ ·σε =





n_P

X

j=1

cjδx_j



∗ρε=

n_P

X

j=1

cjρε(· −xj) in D⁰(Rⁿ). (1.4)

Remark 1.1. Notice that in (1.3) and (1.4) the equality holds inD⁰(Rⁿ) and not only inD⁰(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 (1.5)

for some a∈R+. We denote by Xε(Ω) the set of pairs (σ, u) such that u is as stated above and σ satisfies equation (1.4). 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.

(1.6)

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 (1.3). Given any sequenceε= (ε_i)_i∈Nof positive numbers such thatε_i↓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 andσ=m τH¹

x

^Σ,

+∞, otherwise.

(1.7)

The functionf_a:R+→R+(introduced and studied in AppendixA) is the minimum value of some optimization problem depending onaand on the codimensionn−1 (we notef_a^d, withd=n−kin the general case 1≤k≤ n−1). In particular we prove thatf_a is lower semicontinuous, subadditive, increasing,f_a(0) = 0 and that there exists somec >0 such that

1

c ≤ fa(m) 1 +√

a m ≤c form >0. (1.8)

The Γ-convergence holds for the topology of the weak-∗convergence for the sequence of measures (σ_ε) and for the strongL² convergence for the phase field (u_ε). For a sequence (σ_ε, u_ε) we write (σ_ε, u_ε)→(σ, u) ifσ_ε* σ^∗ and ku_ε−uk_L2 →0. In the sequel we first establish that the sequence of functionals (F_ε,a)_ε is coercive with respect to this topology.

Theorem 1.2. Assume thata >0. For any sequence(σε, uε)⊂ M(Ω,Rⁿ)×L²(Ω)with ε↓0, such that Fε,a(σε, uε; Ω)≤F0<+∞,

there exists σ∈ 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 (σ_ε, u_ε)∈ M(Ω,Rⁿ)×L²(Ω) that converges to (σ, u)∈ M(Ω,Rⁿ)×L²(Ω) asε↓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 ((σε, uε))⊂ M(Ω,Rⁿ)×L²(Ω) such that

(σε, uε)−→^ε↓0 (σ, u) inM(Ω,Rⁿ)×L²(Ω)

and

lim sup

ε↓0

Fε,a(σ_ε, u_ε; Ω)≤ Fa(σ, u; Ω).

As already stated, we only considered the casek= 1 in this introduction. Section4is devoted to the extension of Theorems1.2–1.4in the case where the 1-currents (vector measures) are replaced withk-currents.

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

fa

−→a↓0 c1_(0,+∞) pointwise,

for some c >0. As a consequence (1.7) is an approximation ofcH¹(Σ) for a >0 small and the minimization of (1.6) 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 (1.3). In the casek >1, we obtain a variational approximation of thek-Plateau problem.

1.1. Structure of the paper

In Section2we introduce some notation and recall some useful facts about vector measures and currents, we also anticipate the optimization problem defining the cost functionf_a^d and state some results which are proved in Appendix A. In Section 3 we establish Theorems 1.2–1.4. In Section4 we extend these results to the case 1≤k≤n−1. In Section5we discuss the limita↓0.

2. Preliminaries and notation

The canonical orthonormal basis ofRⁿis denoted by the vectorse1, . . . , en.Lⁿdenotes the Lebesgue measure in Rⁿ and given an integer value kwe denote byωk the measure of the unit ball inR^k, i.e.L^k(B1(0)). For a point x∈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 with M(Ω) the vector space ofRadon measures in Ω and with M(Ω,Rⁿ) =M(Ω)ⁿ the vector space of vector valued measures. For a measureµ∈ M(Ω) we denote by|µ|itstotal variation, in the vector case µ∈ M(Ω,Rⁿ) we writeµ=τ|µ|whereτ is a|µ|-measurale map intoSⁿ⁻¹. We say that a measure issupported on a Borel setE if|µ|(Ω\E) = 0. For an integerk < nwe denote byH^k thek-dimensionalHausdorff measure as in [4]. Given a setE ∈Ω, such that,H^k(E) is finite for somekthe restrictionH^k

x

^Edefines a Radon measure in the space M(Ω). A set E ∈Ω is said to becountably k-rectifiable if up to a H^k negligible set N, 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 formω its comass is defined as

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

Let Dk(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 a k-current M(σ) is the supremum of hσ, ωiamong all k-differential forms with comass bounded by 1. For any k-currentσsuch that bothσ and∂σ are of finite mass we say that σis a normalk-current and we write σ∈Nk(Ω). On the spaceDk(Ω) we can define theflat norm by

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

which metrizes the weak-∗topology on currents on compact subsets ofNk(Ω). By the Radon-Nikodym theorem we can identify ak-currentσwith finite mass with the vector valued measureτ µσ whereµσ is a finite positive valued measure andτ is aµσ-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 be k-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 countably k-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 withR_k(Ω) the space of thesek-rectifiable currents. Among these we name out the subsetP_k(Ω) ofk-rectifiable currents for which Σ is a finite union of polyhedra andθis constant on each of them, these will be called polyhedral chains.

Finally theflat chains F_k(Ω) consist of the closure ofP_k(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 that f 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

Pj→P F(P) : (Pj)j ⊂Pk(Ω) andPj→P

as shown in Section 6 of [23]. Furthermore, in [12] the authors show that iff(t)/t→ ∞ast→0, thenF(σ)<∞ if and only if σis 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 section let us recall a sufficient condition for a flat chain to be rectifiable, proved by White in Corollary 6.1 of [24].

Theorem 2.2. Let σ ∈ F_k(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

^{Σthenσ∈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 Section 2.2.

2.4. Reduced problem results in dimension n−k

This section 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 (σε, uε). We postpone the proofs to Appendix A.3–A.5. We set d=n−k,p > d and considerε to be a sequence such thatε↓0. LetBr(0)⊂R^d be the ball of radiusrcentered in the origin. We consider the functional

Eε,a(ϑ, u;Br) :=

Z

Br

ε^p−d|∇u|^p+(1−u)²

ε^d +u|ϑ|² ε

dx

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

Y_ε,a(m, r,r) =˜

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

f_ε,a^d (m, r,r) =˜ inf

Yε,a(m,r,˜r)

E_ε,a(ϑ, u;B_r). (2.2)

Letf_a^d: [0,+∞)−→R+ be defined as

f_a^d(m) =





 min

ˆ r>0

a m²

ωd ˆr^d + ωdrˆ^d+ (d−1)ωdq_∞^d (0,ˆr)

, form >0,

0, form= 0,

(2.3)

with

q_∞^d (ξ,ˆr) := inf

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 A.1. 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 constant κ:=κ(d, p)such that

f_a^d(m)≥κ for everym >0. (2.6)

Proposition 2.5. For fixedm >0letr_∗be the minimizing radius in the definition off_a^d(m) (2.3). For anyδ >0 and ε small enough there exist a function ϑε=c1B_r∗ε with c >0 such that R

Brϑε =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(σ_ε, u_ε; Ω) ≤ F₀. (3.1)

As a first step we prove:

Lemma 3.1. Assumea >0. There existsC≥0, depending only onΩ,F0 anda such that Z

Ω

|σ_ε| ≤ C, ∀ε. (3.2)

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| ≤µ such 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 (3.2). Letλ∈(0,1] and let Ωλ := {x∈Ω : uε> λ}.

Beingσ_εsquare integrable we identify the measureσ_ε with its density with respect toLⁿ. Therefore splitting the total variation of σ_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

u_ε|σ_ε|² ε

^1/2Z

Ω_λ

ε uε

1/2

.

Sinceλ < uε≤1 on Ωλ andR

Ω_λ(uε|σε|²)/(ε) dxbeing bounded byFε,a(σε, uε) from the previous we get Z

Ω_λ

|σε| ≤ Z

Ω_λ

u_ε|σε|² ε

^1/2r

|Ω|ε

λ ≤

r|Ω|ε F₀

λ .

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

Z

Ω\Ω_λ

|σε| ≤ Z

Ω\Ω_λ

u_ε|σ_ε|²

ε +

Z

Ω\Ω_λ

ε uε

.

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

Ω\Ω_λ

|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

|σε|(Ω)≤F₀

2 + F₀

2a(1−λ)² +

r|Ω|ε F₀

λ .

Asa >0, this yields (3.2).

Step 2.We easily see from R

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

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

* σ∗

follows from (3.2). The properties on the support ofµ, on the divergence of σand 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 assume a≥0 and thatσεis bounded in mass. We show that the limitingσis rectifiable.

Proposition 3.2. Assume a≥0 and that the conclusions of Lemma3.1hold true. There exists a Borel subset Σ with finite length and a Borel measurable functionτ : Σ→Sⁿ⁻¹ such that σ=τ|σ|

x

Σ. Moreover, we have the following estimate,

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

This proposition together with Lemma3.2 and Theorem2.3leads to

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

The latter ensures that the limit couple (σ,1) belongs to X and concludes the proof of Theorem1.2. We now establish Proposition3.2

Sketch of the proof: We first define Σ. Then we show in Lemma 3.5 that for x ∈ Σ, we have lim inf_ε↓0Fε,a(σ_ε, u_ε;B(x, r_j))≥κr_j for a sequence of radiir_j ↓0 andκ >0. The proof of the lemma is based on slicing and on the results of AppendixA. 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(Br(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|(Br/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|(Br/4(x))≤θ|σ|(Br(x)) with r =r_k = 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 everyξ >0 there existsrξ =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 Bj 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.

Sending ξto 0 and thenRto ∞, we obtain|σ|(Γ) = 0.

Set Σ := Σe\Γ, from Lemma3.4, we haveσ=τ|σ|

x

^Σ.Recall thatS ={x1, . . . , xn_P}.

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

ε↓0 Fε,a(σε, uε;B(x, rj)) ≥ √ 2κ rj, where κis the constant of Proposition2.4.

Proof. Letx∈Σ\S. Without loss of generality, we assumex= 0 andτ(x) =e1. Letξ >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 every j≥0,

σ(Br_j)·e1 ≥ (1−ξ)|σ|(Br_j) and |σ|(Br_j/4) ≥ θ|σ|(Br_j). (3.3) Let us fix j ≥0 and set, to simplify the notation, r=r_j and r_∗ =r/√

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

Cr_∗:={x:|x1| ≤r_∗ and |x⁰| ≤r_∗}

so that Cr∗⊂BrandB_r/4⊂C_r∗/2, as shown in Figure1. 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 noteχ_r∗(x⁰) =χ(x⁰/r_∗) and for s∈[−r, r], we set

∀s∈[−r, r], gε(s) := e1· Z

B_r∗⁰

σε(s, x⁰)χr_∗(x⁰) dx⁰.

Since σε is divergence free, e1·σε(·, s) has a meaning on the hyperplane {x1 =s} in the sense of trace, moreover,gεis continuous. Now, let us fix ˆr∈[(1−ξ)r_∗, r_∗] such thatµ({−ˆr,r} ׈ B_r⁰) = 0 (which holds true for a.e. ˆr∈[(1−ξ)r_∗, r_∗]) and let us define the mean value,

¯ gε := 1

2ˆr Z rˆ

−ˆr

gε(s) ds.

From σε

* 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. (3.4)

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

From (3.3), we see thatm >0 forξ small 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 construction χr∗(x⁰)1_[−ˆr,ˆr]≥1B_r/4, using the second inequality of (3.3), we have m ≥ 1

2ˆr(θ−ξ)|σ|(B_r) > 0,

for ξ 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 (3.3) that

|Πσ|(Cr_∗)≤

√ξm

θ−ξ2ˆr. (3.5)

Now, for ε small enough, we have ∇ ·σε = 0 in Cr_∗. Using this, we have for almost every s, 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)∇⁰χr_∗(x⁰) ifh < t, (h−r)ˆ ∇⁰χr∗(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). (3.6)

in theL¹(−ˆr,ˆr) topology. Using (3.5), we see that the above right hand side is bounded byc

√ξ

θ−ξm. Taking into account (3.5) 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

B⁰_r

∗

ε^p−n+1|∇uε|^p+(1−uε)²

εⁿ⁻¹ +uε|σε|² ε

dx⁰ dt

≥ Z ˆr

−ˆr

Z

B⁰_r

∗

ε^p−n+1|∇uε|^p+(1−uε)²

εⁿ⁻¹ +uε|χr∗(x⁰)σε|² ε

dx⁰ dt.

(3.7)

Let us set

ϑ^t_ε(x⁰) :=|χr∗(x⁰)σ_ε(t, x⁰)|.

By construction ϑ^t_ε has 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)⊂B⁰_˜rwith ˜r:=³₄r_∗< r_∗.

By definition of the minimization problem introduced in Section2.4, we have Fε,a(σ_ε, u_ε;B_r) ≥

Z ˆr

−ˆr

inf

(ϑ,u)∈Yε,a( ˜m,r,˜r)

E_ε,a(ϑ, u;B_r)

dt= Z rˆ

−ˆr

f_ε,a( ˜m, r,˜r) dt. (3.8) Taking the infimum limit, by Fatou’s lemma and equation (2.6) of Proposition2.4we get

lim inf

ε↓0 F_ε,a(σ_ε, u_ε;B_r)≥ Z rˆ

−ˆr

lim inf

ε↓0 f_ε,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 Fε,a(σε, uε;Br)≥ √ 2κ r.

The proof of Proposition 3.2is then obtainedviathe Besicovitch covering theorem [15].

3.2. Γ-liminf inequality

In this section we prove the Γ-liminf inequality stated in Theorem1.3.

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 set A⊂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 Ω₀ such 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 Ω0such thath(t,·) restricted to Ω0is the identity map, for anyt∈[0,1]. Let σ_t=h(t,·)]σ, indeed lim inf_t↓0F(σ_t,1)≥ F(σ,1) as σ_t* σ. Further^∗ ∇ ·σ_t=∇ ·σ sinceh(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. (3.9) Let us fix λ≥1 and let us note fa,λ(t) := min(fa(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 (3.9) 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 ([4], Thm. 2.19) to the family B and to the Radon measure H_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).

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

Σf_a(m) dH¹which proves the theorem.

Let us now establish the claim (3.9). Sinceσ is a rectifiable measure, we have forH¹-almost everyx∈Σ, 1

2r Z

ϕ(x+ry) d|σ|(y) −→^r↓0 m(x) Z

R

ϕ(tτ(x)) dt for everyϕ∈C_c(Rⁿ), (3.10)

and

1 2r

Z

B(x,r)∩Σ

|τ(y)−τ(x)| d|σ|(y) −→^r↓0 0. (3.11)

Letx∈Σ\ S be 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 forFε,a(σε, uε;C) whereC is 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 of σ 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

σε(s, x⁰)χ^δ(x⁰/r) dx⁰.

We also introduce the mean value

gε^δ,r := 1 2r

Z r

−r

g^δ,r_ε (s) ds.

From (3.10), we have forr >0 small enough, g^δ,r₀ := 1

2r Z r

−r

e₁· Z

Rⁿ⁻¹

σ_ε(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^δ,r_ε (s) ds ≥ (1−2δ)m. (3.12)

We study the variation of g^δ,r_ε (s). Using∇ ·σ_ε= 0 inC_r^δ, we compute as in the proof of Lemma3.5, g_ε^δ,r(t)−gε^δ,r = 1

2r Z

(−r,r)×Bδr

φt(x⁰, h)·σε(x⁰, h) dx⁰ dh

with

φt(h, x⁰) =

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

Using again the convergence σε

* σ, we deduce∗

g^δ,r_ε (t)−g^δ,rε

−→ε↓0 1 2r

Z

(−r,r)×Bδr

φt(x⁰, h)· dσ(x⁰, h),

inL¹(−r, r). Now, sincee₁· ∇⁰χ^δ ≡0, we deduce from (3.11) that the right hand side goes to 0 asr↓0. Hence, forr >0 small enough,

1 2r

Z

(−r,r)×Bδr

φ_t(x⁰, h)·σ(x⁰, h) dx⁰ dh

≤ δm.

Using (3.12), we conclude that forr >0 small enough and then forε >0 small enough, we have g_ε^δ,r(t) ≥ (1−3δ)m, for a.e.t∈(−r, r).

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

Fε,a(σε, uε;C_r^δ) ≥ 2rf_ε,aⁿ⁻¹((1−3δ)m).

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

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

1 +δ²rand taking the liminf asr↓0, we get lim inf

r↓0

H(B^√_1+δ2r) 2√

1 +δ²r ≥ fa((1−3δ)m)

√1 +δ² . Sending δto 0, we get (3.9) by lower semi-continuity off_a.

3.3. Γ-limsup inequality

Proof of Theorem 1.4. Let us suppose F(σ, u; Ω)<+∞, so that in particular u≡1. From Xia [26], we can assume σ to 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 measure σconcentrated on a segment with constant multiplicity. Then we show how to deal with the case of a polyhedral vector measures.

Step 1.(σconcentrated on a segment.) Assume thatσis supported on the segmentI= [0, L]× {0}and writes as m·e₁H¹

x

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

F(σ,1; Ω) =fa(m)H¹(I) =L fa(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 Proposition2.5withd=n−1. Assumer_∗≥1 and let d(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}.

Setuε(x) =uε(d(x, I)) andσ¹_ε=mH¹

x

^{I∗ρε, whereρε}is the mollifier of equation (1.4). We first construct the vector measures

σ¹_ε=σ¹_εe₁ and σ_ε²(x₁, x⁰) =ϑ_ε(|x⁰|)e₁.

Alternatively, σ_ε²=σ∗ρ˜εfor the choice ˜ρε(x1, x⁰) =χB⁰_r∗ε(x⁰)/ ω_n−1(εr_∗)ⁿ⁻¹. Let us highlight some properties ofσ_ε¹andσ_ε². Both vector measures are radial inx⁰, with an abuse of notation we denoteσ¹_ε(x1, s) =σ¹_ε(x1,|x⁰|).

Since, bothσ_ε¹andσ_ε²are obtained trough convolution it holds supp(σ¹_ε)∪supp(σ_ε²)⊂I_r∗εand they are oriented by the vector e₁therefore|σ¹_ε|=σ¹_ε and|σ²_ε|=ϑ_ε. Furthermore for anyx₁, it holds

Z

{x1}×B_r∗ε⁰

σ¹_ε(x₁, x⁰)−ϑ_ε(x⁰)

dx⁰= 0 (3.13)

We construct σε by interpolating between σ¹_ε and σ²_ε. To this aim consider a cutoff function ζε : R→ R+

satisfying

ζε(t) = 1 fort≤r_∗εor t≥L−r_∗ε,

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







σ³_ε·e1= 0,

σ³_ε·ei(x1, x⁰) =−ζ_ε⁰(x1) x_i

|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 σ_ε² through the (n−1)-dimensional disk {x1} ×B⁰. Forσ³_ε we have the following

∇ ·σ³_ε=−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|)−ϑε(|x⁰|)

=−ζ_ε⁰(x1)

σ¹_ε(x1,|x⁰|)−ϑε(|x⁰|)

. (3.14)

Let

σε=ζεσ_ε¹+ (1−ζε)σ²_ε+σ_ε³.

In force of equation (3.14) and from the construction ofσ¹_ε,σ²_ε andζεwe have

∇ ·σε=∇ ·(ζεσ_ε¹) +∇ ·(1−ζε)σ²_ε+∇ ·σ_ε³=ζε∇ ·σ_ε¹+ζ_ε⁰(σ¹_ε−ϑε) +∇ ·σ_ε³=ζε∇ ·σ_ε¹=∇ ·(σ∗ρε).

In addition for any (x1, x⁰) such that|x⁰| ≥r_∗εfrom (3.13) 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

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

which justifies supp(σε)⊂Ir∗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\I_r, C_r,ε :=I_r∩[2ε, L−2ε]×Rⁿ⁻¹ and D_ε and D_ε⁰, as show in Figure 2, whereD_ε={x1≤2r_∗ε} ∩I_r∗εandD_ε⁰ ={x1≥L−2r_∗ε} ∩I_r∗ε. On Ω\I_rwe notice thatσ_ε= 0 andu_ε= 1 therefore

F_ε,a(σ_ε, u_ε; Ω\I_r) = 0.

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

D_ε

|σε|²dx≤2 m²r²_∗ εⁿ⁻²

Z

B₁

ρ²dx+C

.

Taking into consideration this estimate we obtain

F_ε,a(σ_ε, u_ε;D_ε) =F_ε,a(σ_ε, u_ε;D⁰_ε)≤ (1−η)²

εⁿ⁻¹ Lⁿ(D_ε) + 2m²r_∗² η

εⁿ⁻². (3.15)

Finally on C_r,ε both σ_ε and u_ε are independent of x₁ and are radial in x⁰ then by Fubini’s theorem and Proposition 2.5we get

F_ε,a(σ_ε, u_ε;C_r,ε) =

Z L−2εr∗

2εr∗

Z

B_r⁰

E_ε,a(ϑ_ε, u_ε)≤ L(f_a(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 of σε we need to replace the functionsζε with

ζ˜ε(t) = 1 fort≤εor t≥L−ε,

ζ˜_ε(t) = 0 for 2ε≤t≤L−2ε, and

ζ˜_ε⁰ ≤1

ε.

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

√

3ε centered 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.

This choice ensures that σε 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

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

with these choices foruε andσε the 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 Section2.3it is sufficient to show equation (1.4) for a polyhedral vector measure. Following the same notation introduced therein let

σ=

N

X

j=1

mjH¹

x

^Σjτj.

With no loss of generality we can assume that the segments Σj intersect at most at their extremities. We consider measures σ satisfying constraint (1.3) so that if a point P belongs to Σj₁, . . . ,Σj_P it must satisfy of Kirchhoff law,

j_P

X

j₁

z_jm_j =

(ci, ifP ∈S,

0, otherwise, (3.16)

where zj, is +1 ifP is the ending point of the segment Σj with respect to its orientation, and−1 if it is the starting point. Letσ^j_εand u^j_εbe the sequences constructed above for each segment Ik and define

σε=

N

X

j=1

σ^j_ε and uε= min

j

u^j_ε .