Variational approximation of size-mass energies for k-dimensional currents

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Variational approximation of size-mass energies for k-dimensional currents

Antonin Chambolle, Luca A.D. Ferrari, Benoit Merlet

To cite this version:

Antonin Chambolle, Luca A.D. Ferrari, Benoit Merlet. Variational approximation of size-mass energies

for k-dimensional currents. ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences,

2019, 25, pp.43. �10.1051/cocv/2018027�. �hal-02952640�

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 energies F

ε,a^k

modeled on the Ambrosio-Tortorelli functional. For the choice k = 1 we show that F

ε,a¹

Γ-converges to a branched transportation energy whose cost per unit length is a function f

_aⁿ⁻¹

depending on a parameter a > 0 and on the codimension n − 1. The limit cost f

a

(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 limit a ↓ 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 on k-currents, k < n. The energies F

ε,a^k

then could be used for the numerical treatment of the k-Plateau problem.

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

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

1. Introduction

Let Ω ⊂ R

ⁿ

be a convex, bounded open set. We consider vector measures σ ∈ M(Ω, R

ⁿ

) of the form

σ = m τ H

¹

x ^{Σ, (1.1)}

where Σ is a 1-dimensional rectifiable set oriented by a Borel measurable tangent map τ : Σ → S

ⁿ⁻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) dH

¹

, if σ = (m, τ, Σ), +∞, otherwise in M(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:luca.ferrari@polytechnique.edu

Article published by EDP Sciences ©EDP Sciences, SMAI 2019

Next, given S = {x

1

, . . . , x

n_P

} ⊂ Ω a finite set of points and c

1

, . . . , c

n_P

∈ R such that P

n_P

j=1

c

j

= 0, we consider the optimization problem F(σ) for σ ∈ M(Ω, R

ⁿ

) satisfying

∇ · σ =

n_P

X

j=1

c

j

δ

x_j

in D

⁰

( R

ⁿ

). (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 {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 measure σ = (m, τ, Σ) by a diffuse object (a smooth vector field).

For this, we introduce below a family of corresponding “diffuse” functionals F

ε,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 functionals F

_ε,a

to extend to any ambient dimension n ≥ 2 the phase-field approximation for a branched transportation energy introduced in [10] for n = 2. The approximating functionals are modeled on the ones from Ambrosio and Tortorelli [2] and allow to recover in the limit as ε ↓ 0 an energy of the form (1.2) for a particular choice of the cost function f = f

a

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-vector ξ(x) = ξ

1

(x) ∧ · · · ∧ ξ

k

(x) and m : Σ → R

+

is a Borel measurable function (the multiplicity). The functional (1.2) extends to k-currents σ as follows,

F(σ) :=





 Z

Σ

f (m(x)) dH

^k

, if σ = (m, ξ, Σ),

+∞, otherwise.

We will further show that for small values of a parameter a we can approach solutions to the k-Plateau problem.

Let us define the approximate functionals and describe our main results in the case k = 1. For our phase field approximations we relax the condition on the vector measure σ replacing it by a vector field σ

ε

∈ L

²

(Ω, R

ⁿ

). We then need to mollify condition (1.3). Let ρ : R

ⁿ

→ R

+

be a classical radial mollifier such that supp ρ ⊂ B

1

(0) and R

B₁(0)

ρ = 1. For ε > 0, we set ρ

ε

= ε

⁻ⁿ

ρ(·/ε). We substitute for (1.3) the condition

∇ · σ

ε

=





n_P

X

j=1

c

j

δ

x_j



 ∗ ρ

ε

=

n_P

X

j=1

c

j

ρ

ε

(· − x

j

) in D

⁰

( R

ⁿ

). (1.4)

Remark 1.1. Notice that in (1.3) and (1.4) the equality holds in D

⁰

( R

ⁿ

) and not only in D

⁰

(Ω) so that there is no flux trough ∂Ω.

We also consider the functions u ∈ W

^1,p

(Ω, [η, 1]) such that u ≡ 1 on ∂Ω where η = η(ε) satisfies

η = a ε

ⁿ

(1.5)

for some a ∈ R

+

. We denote by X

ε

(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, R

ⁿ

) × L

²

(Ω). For (σ, u) ∈ M(Ω, R

ⁿ

) × L

²

(Ω) and p > n − 1 we set

F

ε,a

(σ, u; Ω) :=



 

  Z

Ω

ε

^p−n+1

|∇u|

^p

+ (1 − u)

²

ε

ⁿ⁻¹

+ u|σ|

²

ε

dx, if (σ, u) ∈ X

ε

(Ω),

+∞, in the other cases.

(1.6)

Let X be the subset of M(Ω, R

ⁿ

) × L

²

(Ω) consisting of those couples (σ, u) such that u ≡ 1 and σ = (m, τ, Σ) satisfies the constraint (1.3). Given any sequence ε = (ε

_i

)

_i∈N

of positive numbers such that ε

_i

↓ 0, we show that the above family of functionals Γ-converges to

F

a

(σ, u; Ω) =



 

  Z

Σ∩Ω

f

a

(m(x)) dH

¹

(x), if (σ, u) ∈ X and σ = m τ H

¹

x ^Σ,

+∞, otherwise.

(1.7)

The function f

_a

: R

+

→ R

+

(introduced and studied in Appendix A) 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 that f

_a

is lower semicontinuous, subadditive, increasing, f

_a

(0) = 0 and that there exists some c > 0 such that

1

c ≤ f

a

(m) 1 + √

a m ≤ c for m > 0. (1.8)

The Γ-convergence holds for the topology of the weak-∗ convergence for the sequence of measures (σ

_ε

) and for the strong L

²

convergence for the phase field (u

_ε

). For a sequence (σ

_ε

, u

_ε

) we write (σ

_ε

, u

_ε

) → (σ, u) if σ

_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 that a > 0. For any sequence (σ

ε

, u

ε

) ⊂ M(Ω, R

ⁿ

) × L

²

(Ω) with ε ↓ 0, such that F

ε,a

(σ

ε

, u

ε

; Ω) ≤ F

0

< +∞,

there exists σ ∈ M(Ω, R

ⁿ

) 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, R

ⁿ

) × L

²

(Ω) that converges to (σ, u) ∈ M(Ω, R

ⁿ

) × L

²

(Ω) as ε ↓ 0 it holds

lim inf

ε↓0

F

ε,a

(σ

ε

, u

ε

; Ω) ≥ F

a

(σ, u; Ω).

We also establish the corresponding Γ-limsup inequality

Theorem 1.4. Assume that a ≥ 0. For any (σ, u) ∈ M(Ω, R

ⁿ

) × L

²

(Ω) there exists a sequence ((σ

ε

, u

ε

)) ⊂ M(Ω, R

ⁿ

) × L

²

(Ω) such that

(σ

ε

, u

ε

) −→

^ε↓0

(σ, u) in M(Ω, R

ⁿ

) × L

²

(Ω)

and

lim sup

ε↓0

F

ε,a

(σ

_ε

, u

_ε

; Ω) ≤ F

a

(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 1.2–1.4 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 case a > 0. However, as a ↓ 0 we have the important phenomena:

f

a

−→

a↓0

c1

_(0,+∞)

pointwise,

for some c > 0. As a consequence (1.7) is an approximation of cH

¹

(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 case k > 1, we obtain a variational approximation of the k-Plateau problem.

1.1. 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 function f

_a^d

and state some results which are proved in Appendix A. In Section 3 we establish Theorems 1.2–1.4. 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 R

ⁿ

is denoted by the vectors e

1

, . . . , e

n

. L

ⁿ

denotes the Lebesgue measure in R

ⁿ

and given an integer value k we denote by ω

k

the measure of the unit ball in R

^k

, i.e. L

^k

(B

1

(0)). For a point x ∈ R

ⁿ

we note x = (x

1

; x

⁰

) ∈ R × R

ⁿ⁻¹

. For any Borel-measurable set A ⊂ R

ⁿ

we denote by 1

A

(x) the characteristic function of the set A

1

A

(x) :=

( 1 if x ∈ A, 0 otherwise.

Given a vector space Y and its dual Y

⁰

for ω ∈ Y and σ ∈ Y

⁰

we write hω, σi for the dual pairing.

2.1. Measures and vector measures

We denote with M(Ω) the vector space of Radon measures in Ω and with M(Ω, R

ⁿ

) = M(Ω)

ⁿ

the vector

space of vector valued measures. For a measure µ ∈ M(Ω) we denote by |µ| its total variation, in the vector case

µ ∈ M(Ω, R

ⁿ

) we write µ = τ |µ| where τ is a |µ|-measurale map into S

ⁿ⁻¹

. We say that a measure is supported

on a Borel set E if |µ|(Ω \ E) = 0. For an integer k < n we denote by H

^k

the k-dimensional Hausdorff measure

as in [4]. 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 space M(Ω). A set E ∈ Ω is said to be countably k-rectifiable if up to a H

^k

negligible set N , E \ N is

contained in a countable union of C

¹

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 form ω its comass is defined as

kωk = sup{hω, ξi : ξ is a unit, simple k-vector}.

Let D

k

(Ω) be the dual to D

^k

(Ω) i.e. the space of k-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, ωi among all k-differential forms with comass bounded by 1. For any k-current σ such that both σ and ∂σ are of finite mass we say that σ is a normal k-current and we write σ ∈ N

k

(Ω). On the space D

k

(Ω) we can define the flat norm by

F (σ) = inf {M (R) + M (S) : σ = R + ∂S where S ∈ D

k+1

(Ω) and R ∈ D

k

(Ω)} ,

which metrizes the weak-∗ topology on currents on compact subsets of N

k

(Ω). By the Radon-Nikodym theorem we can identify a k-current σ with finite mass with the vector valued measure τ µ

σ

where µ

σ

is a finite positive valued measure and τ is a µ

σ

-measurable map in the set of unitary k-vectors for the mass norm. In particular the action of σ on ω can be written as

hσ, ωi = Z

Ω

hω, τi dµ

σ

.

For a finite mass k-current the mass of σ coincides with the total variation of the measure µ

σ

. A k-current σ is said to be k-rectifiable if we can associate to it a triplet (θ, τ, Σ) such that

hσ, ωi = Z

Σ

θhω, τ i dH

^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 and θ is an L

¹

(Ω, H

^k

x Σ) function that can be assumed positive. We will denote with R

_k

(Ω) the space of these k-rectifiable currents. Among these we name out the subset P

_k

(Ω) of k-rectifiable currents for which Σ is a finite union of polyhedra and θ is constant on each of them, these will be called polyhedral chains.

Finally the flat chains F

_k

(Ω) consist of the closure of P

_k

(Ω) in the weak-∗ topology. By the scheme of Federer ([16], 4.1.24) it holds

P

k

(Ω) ⊂ N

k

(Ω) ⊂ F

k

(Ω).

Remark 2.1 (1-Currents and Vector Measures). Since the vector spaces Λ

1

R

ⁿ

, Λ

¹

R

ⁿ

identify with R

ⁿ

, any

vector measure σ ∈ M(Ω, R

ⁿ

) 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, R

ⁿ

) 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

For f : R 7→ R

⁺

an even function we define a functional P

k

(Ω) −→ R

+

, P = X

j

(m

j

, τ

j

, Σ

j

) 7−→ F (P) = X

j

f (m

j

)H

^k

(Σ

j

),

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

F

k

(Ω) −→ R

+

,

P 7−→ F (P) = inf

lim inf

Pj→P

F(P ) : (P

j

)

j

⊂ P

k

(Ω) and P

j

→ P

as shown in Section 6 of [23]. Furthermore, in [12] the authors show that if f (t)/t → ∞ as t → 0, then F (σ) < ∞ if and only if σ is rectifiable and for any such σ the functional takes the explicit form

F(σ) = Z

Σ

f(m(x)) dH

^k

(x) if σ = (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 that σ = σ x ^{Σ then σ ∈ R}

^k

^{(Ω) i.e., σ writes as (m, τ, Σ).}

In the context of vector measures the theorem writes as

Theorem 2.3. Let σ ∈ M(Ω, R

ⁿ

). If |σ|(Ω) + |∇ · σ|(Ω) < ∞, ∇ · σ is at most a countable sum of Dirac masses and there exists a Borel set Σ with H

¹

(Σ) < ∞ and σ = σ x ^{Σ then σ} is 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 (σ

ε

, u

ε

). We postpone the proofs to Appendix A.3–A.5. We set d = n − k, p > d and consider ε to be a sequence such that ε ↓ 0. Let B

r

(0) ⊂ R

^d

be the ball of radius r centered in the origin. We consider the functional

E

ε,a

(ϑ, u; B

r

) :=

Z

Br

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 ∈ L

²

(B

r

) is such that supp(ϑ) ⊂ B

r˜

with 0 < r < r, ˜ kϑk

1

= m. This leads to define the set

Y

_ε,a

(m, r, r) = ˜

(ϑ, u) ∈ L

²

(B

_r

) × W

^1,p

(B

_r

, [η, 1]) : kϑk

₁

= m and 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)

Let f

_a^d

: [0, +∞) −→ R

+

be defined as

f

_a^d

(m) =



 

  min

ˆ r>0

a m

²

ω

d

ˆ r

^d

+ ω

d

r ˆ

^d

+ (d − 1) ω

d

q

_∞^d

(0, ˆ r)

, for m > 0,

0, for m = 0,

(2.3)

with

q

_∞^d

(ξ, ˆ r) := inf

Z

+∞

ˆ r

t

^d−1

|v

⁰

|

^p

+ (1 − v)

²

dt : v(ˆ r) = ξ and lim

t→+∞

v(t) = 1

, (2.4)

for ˆ r > 0, ξ ≥ 0. For a graph of the profile v realizing the infimum in the latter see Figure A.1. We have the following results.

Proposition 2.4. For any r > 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 every m > 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

B_r∗ε

with c > 0 such that R

Br

ϑ

ε

= m and a nondecreasing radial function u

ε

: B

r

7→ [η, 1] such that u

ε

(0) = η, u

ε

= 1 on ∂B

r

and

E

_ε,a

(ϑ

_ε

, u

_ε

; B

_r

) ≤ f

_a^d

(m) + δ. (2.7) Proposition 2.6. The function f

_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 (F

ε,a

)

ε

. Let us consider a family of functions (σ

ε

, u

ε

)

_ε↓0

, such that (σ

ε

, u

ε

) ∈ X

ε

(Ω) and

F

ε,a

(σ

_ε

, u

_ε

; Ω) ≤ F

₀

. (3.1)

As a first step we prove:

Lemma 3.1. Assume a > 0. There exists C ≥ 0, depending only on Ω, F

0

and a such that Z

Ω

|σ

_ε

| ≤ C, ∀ ε. (3.2)

As a consequence there exist a positive Radon measure µ ∈ ( R

ⁿ

, R

+

) supported in Ω and a vectorial Radon measure σ ∈ M(Ω, R

ⁿ

) with ∇ · σ = P a

_j

δ

_xj

and |σ| ≤ µ such that up to a subsequence

u

ε

→ 1 in L

²

(Ω), |σ

ε

| * µ

^∗

in M( R

ⁿ

), σ

ε

* σ

∗

in M( R

ⁿ

, R

ⁿ

).

Proof. We divide the proof into three steps.

Step 1. We start by proving the uniform bound (3.2). Let λ ∈ (0, 1] and let Ω

λ

:= {x ∈ Ω : u

ε

> λ} .

Being σ

_ε

square integrable we identify the measure σ

_ε

with its density with respect to L

ⁿ

. Therefore splitting the total variation of σ

_ε

, we write

|σ

ε

|(Ω) = Z

Ω

|σ

ε

| dx = Z

Ωλ

|σ

ε

| dx + Z

Ω\Ωλ

|σ

ε

| dx.

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

Ω_λ

|σ

ε

| ≤ Z

Ω_λ

u

_ε

|σ

_ε

|

²

ε

^1/2

Z

Ω_λ

ε u

ε

1/2

.

Since λ < u

ε

≤ 1 on Ω

λ

and R

Ω_λ

(u

ε

|σ

ε

|

²

)/(ε) dx being bounded by F

ε,a

(σ

ε

, u

ε

) from the previous we get Z

Ω_λ

|σ

ε

| ≤ Z

Ω_λ

u

_ε

|σ

ε

|

²

ε

^1/2

r

|Ω|ε

λ ≤

r |Ω| ε F

₀

λ .

Next, in Ω \ Ω

λ

, by Young inequality, we have 2

Z

Ω\Ω_λ

|σ

ε

| ≤ Z

Ω\Ω_λ

u

_ε

|σ

_ε

|

²

ε +

Z

Ω\Ω_λ

ε u

ε

.

Using u

_ε

≥ η(ε), η/ε

ⁿ

= a and (1 − λ)

²

≤ (1 − u

_ε

)

²

in Ω \ Ω

_λ

, we obtain Z

Ω\Ω_λ

|σ

ε

| ≤ 1 2

Z

Ω

u

ε

|σ

ε

|

²

ε + ε

ⁿ

2 η (1 − λ)

²

Z

Ω

(1 − u

ε

)

²

ε

ⁿ⁻¹

≤ F

0

2 + F

0

2 a (1 − λ)

²

. Hence

|σ

ε

|(Ω) ≤ F

₀

2 + F

₀

2 a (1 − λ)

²

+

r |Ω| ε F

₀

λ .

As a > 0, this yields (3.2).

Step 2. We easily see from R

Ω

(1 − u

_ε

)

²

≤ F

₀

ε

ⁿ⁻¹

that u

_ε

→ 1 in L

²

(Ω) as ε ↓ 0.

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

ε

| * µ

^∗

and σ

ε

* σ

∗

follows from (3.2). The properties on the support of µ, on the divergence of σ and the fact that |σ| ≤ µ follow from the respective properties of σ

ε

.

We have just showed that the limit σ of a family (σ

ε

, u

ε

)

ε

equibounded in energy is bounded in mass. In

what follows, we assume a ≥ 0 and that σ

ε

is bounded in mass. We show that the limiting σ is rectifiable.

Proposition 3.2. Assume a ≥ 0 and that the conclusions of Lemma 3.1 hold true. There exists a Borel subset Σ with finite length and a Borel measurable function τ : Σ → S

ⁿ⁻¹

such that σ = τ|σ| x Σ. Moreover, we have the following estimate,

H

¹

(Σ) ≤ C

_∗

F

₀

, where the constant C

_∗

≥ 0 only depends on d and p.

This proposition together with Lemma 3.2 and Theorem 2.3 leads to

Proposition 3.3. σ is a 1-rectifiable vector measure and in particular Σ is a countably H

¹

-rectifiable set.

The latter ensures that the limit couple (σ, 1) belongs to X 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 for x ∈ Σ, we have lim inf

_ε↓0

F

ε,a

(σ

_ε

, u

_ε

; B(x, r

_j

)) ≥ κr

_j

for a sequence of radii r

_j

↓ 0 and κ > 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|(B

r

(x)) > 0 and ∃τ = τ(x) ∈ S

ⁿ⁻¹

such that τ = lim

r↓0

σ(B

r

(x))

|σ|(B

r

(x))

.

We observe that by Besicovitch derivation theorem,

σ = τ|σ| x ^{Σ. e}

Next we fix θ ∈ (0, 1/4

ⁿ

) and define Γ :=

x ∈ Σ : e ∃ r

0

> 0 such that, |σ|(B

r/4

(x))

|σ|(B

r

(x)) ≤ θ for every r ∈ (0, r

0

]

.

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|(B

r

(x)) with r = r

_k

= 4

^−k

r

₀

, k ≥ 0, we get

|σ|(B

r_k

) ≤ θ

^k

|σ|(B

r₀

). Hence there exists C ≥ 0 such that

|σ|(B

r

(x)) ≤ Cr

(ln 1/θ)/(ln 4)

.

Letting λ = (ln

¹_θ

)/(ln 4), we have by assumption λ > n. Therefore, for every ξ > 0 there exists r

ξ

= r

ξ

(x) ∈ (0, 1) such that

|σ|(B

r_ξ

(x)) ≤ ξ|B

r_ξ

(x)|.

Now, for R > 0, we cover Γ ∩ B

R

with balls of the form B

_rξ(x)

(x). Using Besicovitch covering theorem, we have

Γ ∩ B

R

⊂ ∪

^N(n)_j=1

B

j

,

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)

(x

k

). 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 ξ to 0 and then R to ∞, we obtain |σ|(Γ) = 0.

Set Σ := Σ e \ Γ, from Lemma 3.4, we have σ = τ|σ| x ^Σ. Recall that S = {x

1

, . . . , x

n_P

}.

Lemma 3.5. For every x ∈ Σ \ S , there exists a sequence (r

j

) = (r

j

(x)) ⊂ (0, 1) with r

j

↓ 0 such that lim inf

ε↓0

F

ε,a

(σ

ε

, u

ε

; B(x, r

j

)) ≥ √ 2 κ r

j

, where κ is the constant of Proposition 2.4.

Proof. Let x ∈ Σ \ S . Without loss of generality, we assume x = 0 and τ (x) = e

1

. Let ξ > 0 be a small parameter to be fixed later. From the definition of Σ, there exists a sequence (r

j

) = (r

j

(x)) ⊂ (0, d(x, S )) such that for every j ≥ 0,

σ(B

r_j

) · e

1

≥ (1 − ξ)|σ|(B

r_j

) and |σ|(B

r_j/4

) ≥ θ|σ|(B

r_j

). (3.3) Let us fix j ≥ 0 and set, to simplify the notation, r = r

_j

and r

_∗

= r/ √

2. Recall the notation x = (x

₁

, x

⁰

) ∈ R × R

ⁿ⁻¹

and define the cylinder

C

r_∗

:= {x : |x

1

| ≤ r

_∗

and |x

⁰

| ≤ r

_∗

}

so that C

r∗

⊂ B

r

and B

_r/4

⊂ C

_r∗/2

, as shown in Figure 1. Let χ ∈ C

_c^∞

( R

ⁿ⁻¹

, [0, 1]) be a radial cut-off function such that χ(x

⁰

) = 1 if |x

⁰

| ≤

¹₂

and χ(x

⁰

) = 0 for |x

⁰

| ≥

³₄

. Then, we note χ

_r∗

(x

⁰

) = χ(x

⁰

/r

_∗

) and for s ∈ [−r, r], we set

∀s ∈ [−r, r], g

ε

(s) := e

1

· Z

B_r∗⁰

σ

ε

(s, x

⁰

) χ

r_∗

(x

⁰

) dx

⁰

.

Since σ

ε

is divergence free, e

1

· σ

ε

(·, s) has a meaning on the hyperplane {x

1

= s} in the sense of trace, moreover, g

ε

is continuous. Now, let us fix ˆ r ∈ [(1 − ξ)r

_∗

, r

_∗

] such that µ({−ˆ r, r} × ˆ B

_r⁰

) = 0 (which holds true for a.e. ˆ r ∈ [(1 − ξ)r

_∗

, r

_∗

]) and let us define the mean value,

¯ g

ε

:= 1

2ˆ r Z

rˆ

−ˆr

g

ε

(s) ds.

From σ

ε

* σ,

∗

|σ

ε

| * µ, we have

^∗

lim

ε↓0

¯ g

_ε

= 1 2ˆ r

Z

(−ˆr,ˆr)×B_r∗⁰

χ

_r∗

(x

⁰

) dσ(s, x

⁰

)

!

· e

₁

=: m. (3.4)

Figure 1. Illustration of the sections of B

_r

, B

_r/4

and C

_r∗

. In grayscale we represent the level sets of the function χ

_r∗

(x

⁰

)1

_[−ˆr,ˆr]

.

From (3.3), we see that m > 0 for ξ small enough. Indeed, we have (1 − ξ)|σ|(B

r

) ≤ +2ˆ rm + σ(B

r

) · e

1

=

Z

Br

1 − χ

r∗

(x

⁰

)1

_[−ˆr,ˆr]

dσ(s, x

⁰

) · e

1

≤ 2ˆ rm + Z

B_r

1 − χ

_r∗

(x

⁰

)1

_[−ˆr,ˆr]

d|σ|(s, x

⁰

)

≤ 2ˆ rm + |σ|(B

r

) − Z

Br

χ

r∗

(x

⁰

)1

_[−ˆr,ˆr]

d|σ|(s, x

⁰

).

Since by construction χ

r∗

(x

⁰

)1

_[−ˆr,ˆr]

≥ 1

B_r/4

, using the second inequality of (3.3), we have m ≥ 1

2ˆ r (θ − ξ)|σ|(B

_r

) > 0,

for ξ small enough. Similarly, denoting Π : R

ⁿ

→ R

ⁿ⁻¹

, (t, x

⁰

) 7→ x

⁰

the orthogonal projection onto the last (n − 1) coordinates, we deduce again from (3.3) that

|Πσ|(C

r_∗

) ≤

√ ξm

θ − ξ 2ˆ r. (3.5)

Now, for ε small enough, we have ∇ · σ

ε

= 0 in C

r_∗

. Using this, we have for almost every s, t ∈ [−ˆ r, r], with ˆ s < t,

g

ε

(t) − g

ε

(s) = Z

t

s

"

Z

B_r∗⁰

σ

ε

(x

⁰

, h) · ∇

⁰

χ

r∗

(x

⁰

) dx

⁰

# dh.

Integrating in s over (−ˆ r, r), we get for almost every ˆ t ∈ [−r, r], g

ε

(t) − ¯ g

ε

= 1

2ˆ r Z

(−ˆr,ˆr)×B_r∗⁰

φ

t

(x

⁰

, h) · σ

ε

(x

⁰

, h) dx

⁰

dh

with

φ

_t

(h, x

⁰

) =

( (h + ˆ r) ∇

⁰

χ

r_∗

(x

⁰

) if h < t, (h − r) ˆ ∇

⁰

χ

r∗

(x

⁰

) if h > t.

We deduce the following convergence

g

ε

(t) − m −→

^ε↓0

1 2ˆ r

Z

(−ˆr,ˆr)×B_r∗⁰

φ

t

(h, x

⁰

) · dσ(h, x

⁰

). (3.6)

in the L

¹

(−ˆ r, ˆ r) topology. Using (3.5), we see that the above right hand side is bounded by c

√ξ

θ−ξ

m. Taking into account (3.5) and the continuity of g

ε

, we conclude that

lim inf

ε↓0

g

_ε

(t) ≥

1 − c

√ ξ θ − ξ

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

F

ε,a

(σ

ε

, u

ε

; B

r

) ≥ Z

ˆr

−ˆr

Z

B⁰_r

∗

ε

^p−n+1

|∇u

ε

|

^p

+ (1 − u

ε

)

²

ε

ⁿ⁻¹

+ u

ε

|σ

ε

|

²

ε

dx

⁰

dt

≥ Z

ˆr

−ˆr

Z

B⁰_r

∗

ε

^p−n+1

|∇u

ε

|

^p

+ (1 − u

ε

)

²

ε

ⁿ⁻¹

+ u

ε

|χ

r∗

(x

⁰

)σ

ε

|

²

ε

dx

⁰

dt.

(3.7)

Let us set

ϑ

^t_ε

(x

⁰

) := |χ

r∗

(x

⁰

)σ

_ε

(t, x

⁰

)|.

By construction ϑ

^t_ε

has the properties:

– ϑ

^t_ε

∈ L

¹

(B

_r⁰_∗

), – lim inf

_ε↓0

R

B_r∗⁰

ϑ

^t_ε

(x

⁰

) dx

⁰

≥ lim inf

_ε↓0

g

_ε

(t) ≥ 1 − c

√ξ θ−ξ

m = ˜ m > 0, – supp(ϑ

^t_ε

) ⊂ B

⁰_˜r

with ˜ r :=

³₄

r

_∗

< r

_∗

.

By definition of the minimization problem introduced in Section 2.4, we have F

ε,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 Proposition 2.4 we 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 ˆ r κ.

The latter holds for almost every ˆ r ∈ [(1 − ξ)r

_∗

, r

_∗

] and eventually, since the r

_∗

= r/ √

2, we conclude lim inf

ε↓0

F

ε,a

(σ

ε

, u

ε

; B

r

) ≥ √

2 κ r.

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

3.2. Γ-liminf inequality

In this section we prove the Γ-liminf inequality stated in Theorem 1.3.

Proof of Theorem 1.3. With no loss of generality we assume that lim inf

_ε↓0

F

ε,a

(σ

ε

, u

ε

) < +∞ otherwise the inequality is trivial. For a Borel set A ⊂ Ω, we define

H (A) := lim inf

ε↓0

F

ε,a

(σ

ε

, u

ε

; A),

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

¹

x Σ. Furthermore we can assume σ to be compactly supported in Ω. Consider a convex open set Ω

₀

such that supp(∇ · σ) = S ⊂⊂ Ω

₀

⊂⊂ Ω and let h := [0, 1] × R

ⁿ

→ R

ⁿ

be a smooth homotopy of the indentity map on R

ⁿ

onto a contraction of Ω into Ω

0

such that h(t, ·) restricted to Ω

0

is the identity map, for any t ∈ [0, 1]. Let σ

_t

= h(t, ·)]σ, indeed lim inf

_t↓0

F (σ

_t

, 1) ≥ F (σ, 1) as σ

_t

* σ. Further

^∗

∇ · σ

_t

= ∇ · σ since h(t, ·) is the identity on S . Now we claim that

lim inf

r↓0

H

B(x, r)

2r ≥ f

a

(m(x)) for H

¹

-almost every x ∈ Σ. (3.9) Let us fix λ ≥ 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) dH

¹

.

Now, let δ ∈ (0, 1). Assuming that (3.9) holds true, there exists Σ

⁰

⊂ Σ with H

¹

(Σ\Σ

0

) = 0 such that for every x ∈ Σ

0

, there exists r

0

(x) > 0 with

(1 + δ)H

B(x, r)

≥ 2rf

_a,λ

(m(x)) for every r ∈ (0, r

₀

(x)).

By the Besicovitch differentiation Theorem, there exists Σ

1

⊂ Σ with H

¹

(Σ\Σ

1

) = 0 such that for every x ∈ Σ

1

, there exists r

1

(x) > 0 with

(1 + δ)2rf

a

(m(x)) ≥ H

_λ⁰

B(x, r)

for every r ∈ (0, r

1

(x)).

We consider the familly B of closed balls B(x, r) with x ∈ Σ

₀

∩ Σ

₁

and 0 < r < min(r

₀

(x), r

₁

(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 balls B

⁰

⊂ B such 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)

²

H (Ω).

Sending λ to infinity and then δ to 0, we get the lower bound H (Ω) ≥ R

Σ

f

_a

(m) dH

¹

which proves the theorem.

Let us now establish the claim (3.9). Since σ is a rectifiable measure, we have for H

¹

-almost every x ∈ Σ, 1

2r Z

ϕ(x + ry) d|σ|(y) −→

^r↓0

m(x) Z

R

ϕ(tτ (x)) dt for every ϕ ∈ C

_c

( R

ⁿ

), (3.10)

and

1 2r

Z

B(x,r)∩Σ

|τ(y) − τ(x)| d|σ|(y) −→

^r↓0

0. (3.11)

Let x ∈ Σ \ S be such a point. Without loss of generality, we assume x = 0, τ (0) = e

1

and m := m(0) > 0. Let δ ∈ (0, 1). Our goal is to establish a precise lower bound for F

ε,a

(σ

ε

, u

ε

; C) where C is a cylinder of the form

C

_r^δ

:= {x ∈ R

ⁿ

: |x

1

| < δr, |x

⁰

| < 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^∞

( R

ⁿ⁻¹

, [0, 1]) be a radial cut-off function with χ

^δ

(x

⁰

) = 1 if |x

⁰

| ≤ δ/2, χ

^δ

(x

⁰

) = 0 if |x

⁰

| ≥ δ. For ε > 0 and r ∈ (0, d(0, ∂Ω)), we define for s ∈ (−r, r),

g

^δ,r_ε

(s) := e

1

· Z

Rⁿ⁻¹

σ

ε

(s, x

⁰

) χ

^δ

(x

⁰

/r) dx

⁰

.

We also introduce the mean value

g

ε^δ,r

:= 1 2r

Z

r

−r

g

^δ,r_ε

(s) ds.

From (3.10), we have for r > 0 small enough, g

^δ,r₀

:= 1

2r Z

r

−r

e

₁

· Z

Rⁿ⁻¹

σ

_ε

(s, x

⁰

) χ

^δ

(x

⁰

/r) dx ds ≥ (1 − δ)m.

For such r > 0, we deduce from σ

ε

* σ

∗

that for ε > 0 small enough

g

ε^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 in C

_r^δ

, we compute as in the proof of Lemma 3.5, g

_ε^δ,r

(t) − g

ε^δ,r

= 1

2r Z

(−r,r)×Bδr

φ

t

(x

⁰

, h) · σ

ε

(x

⁰

, h) dx

⁰

dh

with

φ

t

(h, x

⁰

) =

( (h + ˆ r) ∇

⁰

χ

^δ

(x

⁰

/r) if h < t, (h − ˆ r) ∇

⁰

χ

^δ

(x

⁰

/r) if h > t.

Using again the convergence σ

ε

* σ, we deduce

∗

g

^δ,r_ε

(t) − g

^δ,rε

−→

ε↓0

1 2r

Z

(−r,r)×Bδr

φ

t

(x

⁰

, h) · dσ(x

⁰

, h),

in L

¹

(−r, r). Now, since e

₁

· ∇

⁰

χ

^δ

≡ 0, we deduce from (3.11) that the right hand side goes to 0 as r ↓ 0. Hence, for r > 0 small enough,

1 2r

Z

(−r,r)×Bδr

φ

_t

(x

⁰

, h) · σ(x

⁰

, h) dx

⁰

dh

≤ δm.

Using (3.12), we conclude that for r > 0 small enough and then for ε > 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

F

ε,a

(σ

ε

, u

ε

; C

_r^δ

) ≥ 2rf

_ε,aⁿ⁻¹

((1 − 3δ)m) .

Sending ε ↓ 0 and recalling that we omit the superscript when it is n − 1, we obtain H(C

_r^δ

) ≥ 2rf

a

((1 − 3δ)m) .

We notice that H(B

^√_1+δ2r

) ≥ H (C

_r^δ

). Dividing by 2 √

1 + δ

²

r and taking the liminf as r ↓ 0, we get lim inf

r↓0

H(B

^√_1+δ2r

) 2 √

1 + δ

²

r ≥ f

a

((1 − 3δ)m)

√ 1 + δ

²

. Sending δ to 0, we get (3.9) by lower semi-continuity of f

_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 σ concentrated 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 σ is supported on the segment I = [0, L] × {0} and writes as m · e

₁

H

¹

x

^I

^{. Consider m} constant so that ∇ · σ = m(δ

_(0,0)

− δ

_(L,0)

) and

F(σ, 1; Ω) = f

a

(m) H

¹

(I) = L f

a

(m).

For δ > 0 fixed, we consider the profiles

u

ε

(t) :=



 

 

 

 

η, for 0 ≤ t ≤ r

∗

ε, v

δ

t ε

, for r

∗

ε ≤ t ≤ r,

1 for r ≤ t,

and ϑ

ε

= m χ

B⁰_r∗ε

(x

⁰

) ω

_n−1

(εr

_∗

)

ⁿ⁻¹

with r

_∗

and v

δ

, defined in Proposition 2.5 with d = n − 1. Assume r

_∗

≥ 1 and let d(x, I) be the distance function from the segment I and introduce the sets

I

r∗ε

:= {x ∈ Ω : d(x, I) ≤ r

∗

ε} , and I

r

:= {x ∈ Ω : d(x, I) ≤ r} .

Set u

ε

(x) = u

ε

(d(x, I)) and σ

¹_ε

= mH

¹

x ^{I ∗ ρ}

^ε

^{, where ρ}

^ε

is the mollifier of equation (1.4). We first construct the vector measures

σ

¹_ε

= σ

¹_ε

e

₁

and σ

_ε²

(x

₁

, x

⁰

) = ϑ

_ε

(|x

⁰

|) e

₁

.

Alternatively, σ

_ε²

= σ ∗ ρ ˜

ε

for the choice ˜ ρ

ε

(x

1

, x

⁰

) = χ

B⁰_r∗ε

(x

⁰

)/ ω

_n−1

(εr

_∗

)

ⁿ⁻¹

. Let us highlight some properties of σ

_ε¹

and σ

_ε²

. Both vector measures are radial in x

⁰

, with an abuse of notation we denote σ

¹_ε

(x

1

, s) = σ

¹_ε

(x

1

, |x

⁰

|).

Since, both σ

_ε¹

and σ

_ε²

are obtained trough convolution it holds supp(σ

¹_ε

) ∪ supp(σ

_ε²

) ⊂ I

_r∗ε

and they are oriented by the vector e

₁

therefore |σ

¹_ε

| = σ

¹_ε

and |σ

²_ε

| = ϑ

_ε

. Furthermore for any x

₁

, it holds

Z

{x1}×B_r∗ε⁰

σ

¹_ε

(x

₁

, x

⁰

) − ϑ

_ε

(x

⁰

)

dx

⁰

= 0 (3.13)

We construct σ

ε

by interpolating between σ

¹_ε

and σ

²_ε

. To this aim consider a cutoff function ζ

ε

: R → R

+

satisfying

ζ

ε

(t) = 1 for t ≤ r

_∗

ε or t ≥ L − r

_∗

ε,

ζ

_ε

(t) = 0 for 2 r

_∗

ε ≤ t ≤ L − 2 r

_∗

ε, and |ζ

_ε⁰

| ≤ 1 r

∗

ε . and set



 

 

σ

³_ε

· e

1

= 0,

σ

³_ε

· e

i

(x

1

, x

⁰

) = −ζ

_ε⁰

(x

1

) x

_i

|x

⁰

|

ⁿ⁻¹

Z

|x⁰|

0

s

ⁿ⁻²

σ

¹_ε

(x

1

, s) − ϑ

ε

(s)

ds, for i = 2, . . . , n.

The integral corresponds to the difference of the fluxes of σ

_ε¹

and σ

_ε²

through the (n − 1)-dimensional disk {x

1

} × B

⁰

. For σ

³_ε

we have the following

∇ · σ

³_ε

= − ζ

_ε⁰

(x

1

)

n

X

i=2

" 1

|x

⁰

|

ⁿ⁻¹

− (n − 1)x

²_i

|x

⁰

|

ⁿ⁺¹

Z

|x⁰| 0

s

ⁿ⁻²

σ

¹_ε

(x

1

, s) − ϑ

ε

(s) ds + x

²_i

|x

⁰

|

²

σ

¹_ε

(x

1

, |x

⁰

|) − ϑ

ε

(|x

⁰

|)

= −ζ

_ε⁰

(x

1

)

σ

¹_ε

(x

1

, |x

⁰

|) − ϑ

ε

(|x

⁰

|)

. (3.14)

Let

σ

ε

= ζ

ε

σ

_ε¹

+ (1 − ζ

ε

) σ

²_ε

+ σ

_ε³

.

In force of equation (3.14) and from the construction of σ

¹_ε

, σ

²_ε

and ζ

ε

we have

∇ · σ

ε

= ∇ · (ζ

ε

σ

_ε¹

) + ∇ · (1 − ζ

ε

)σ

²_ε

+ ∇ · σ

_ε³

= ζ

ε

∇ · σ

_ε¹

+ ζ

_ε⁰

(σ

¹_ε

− ϑ

ε

) + ∇ · σ

_ε³

= ζ

ε

∇ · σ

_ε¹

= ∇ · (σ ∗ ρ

ε

).

In addition for any (x

1

, x

⁰

) such that |x

⁰

| ≥ r

_∗

ε from (3.13) we derive

σ

_ε³

· e

i

(x

1

, x

⁰

) = −ζ

_ε⁰

(x

1

) x

i

|x

⁰

|

ⁿ⁻²

Z

|x⁰|

0

s

ⁿ⁻¹

σ

¹_ε

(x

1

, s) − ϑ

ε

(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 ζ

ε

, whilst the striped region corresponds to the cylinder C

_r,ε

.

which justifies supp(σ

ε

) ⊂ I

r∗ε

. Let us now prove lim sup

ε↓0

F

ε,a

(σ

ε

, u

ε

; Ω) ≤ L f

a

(m) + Cδ.

We split Ω as the union of Ω \ I

_r

, C

_r,ε

:= I

_r

∩ [2 ε, L − 2 ε] × R

ⁿ⁻¹

and D

_ε

and D

_ε⁰

, as show in Figure 2, where D

_ε

= {x

1

≤ 2 r

_∗

ε} ∩ I

_r∗ε

and D

_ε⁰

= {x

1

≥ L − 2 r

_∗

ε} ∩ I

_r∗ε

. On Ω \ I

_r

we notice that σ

_ε

= 0 and u

_ε

= 1 therefore

F

_ε,a

(σ

_ε

, u

_ε

; Ω \ I

_r

) = 0.

Observe that |D

ε

| = |D

⁰_ε

| = Cε

ⁿ

, then we have the upper bound Z

D_ε

|σ

ε

|

²

dx ≤ 2 m

²

r

²_∗

ε

ⁿ⁻²

Z

B₁

ρ

²

dx + C

.

Taking into consideration this estimate we obtain

F

_ε,a

(σ

_ε

, u

_ε

; D

_ε

) = F

_ε,a

(σ

_ε

, u

_ε

; D

⁰_ε

) ≤ (1 − η)

²

ε

ⁿ⁻¹

L

ⁿ

(D

_ε

) + 2 m

²

r

_∗²

η

ε

ⁿ⁻²

. (3.15)

Finally on C

_r,ε

both σ

_ε

and u

_ε

are independent of x

₁

and are radial in x

⁰

then by Fubini’s theorem and Proposition 2.5 we 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 case r

∗

< 1. From the point of view of the construction of σ

ε

we need to replace the functions ζ

ε

with

ζ ˜

ε

(t) = 1 for t ≤ ε or t ≥ L − ε,

ζ ˜

_ε

(t) = 0 for 2 ε ≤ t ≤ L − 2 ε, and

ζ ˜

_ε⁰

≤ 1

ε .

Figure 3. On the left the striped region corresponds to supp(σ

ε

), 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 ˜ ζ

ε

in grayscale.

This choice ensures that σ

ε

has all the properties previously obtained with r

_∗

ε replaced by ε accordingly.

Define

w

_ε

(t) :=







η, for t ≤ √

3ε 1 − η

r − √

3 (t − √

3) + η, for √

3ε ≤ t ≤ r.

and set

u

ε

= min{u

ε

(d(x, I)), w

ε

(|x|), w

ε

(|x − (L; 0)|)}.

with these choices for u

ε

and σ

ε

the estimates follow analogously with small differences in the constants.

Step 2. Case of a generic σ in polyhedral form. Indeed, in force of the results quoted in Section 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

j

H

¹

x ^Σ

^j

^τ

^j

^.

With no loss of generality we can assume that the segments Σ

j

intersect at most at their extremities. We consider measures σ satisfying constraint (1.3) so that if a point P belongs to Σ

j₁

, . . . , Σ

j_P

it must satisfy of Kirchhoff law,

j_P

X

j₁

z

_j

m

_j

=

( c

i

, if P ∈ S ,

0, otherwise, (3.16)

where z

j

, is +1 if P is the ending point of the segment Σ

j

with respect to its orientation, and −1 if it is the starting point. Let σ

^j_ε

and u

^j_ε

be the sequences constructed above for each segment I

k

and define

σ

ε

=

N

X

j=1

σ

^j_ε

and u

ε

= min

j

u

^j_ε

.

Let P

j

and Q

j

be respectively the initial and final point of the segment Σ

j

and recall that, by the construction made above, for each j

∇ · σ

^j_ε

= m

_j

δ

_Pj

− δ

_Qj

∗ ρ

_ε

then by linearity of the divergence operator, it holds

∇ · σ

ε

=

N

X

j=1

∇ · σ

_ε^k

=

N

X

j=1

m

j

δ

P_j

− δ

Q_j

∗ ρ

ε

and the latter satisfies constraint (1.4) in force of equation (3.16). To conclude let us prove that

lim sup

ε↓0

F

ε,a

(σ

ε

, u

ε

; Ω) ≤

N

X

j=1

f

a

(m

j

)H

¹

(Σ

j

). (3.17)

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

^j_ε²

) ∩ · · · ∩ supp σ

_ε^jP

6= ∅

for some j

₁

, . . . , j

_P

and all ε. Let r

^j_∗¹

, . . . , r

^j_∗^P

be the radii introduced above for each of these measures, let r

_∗

= max{r

_∗^j1

, . . . , r

_∗^jP

, 1} , set m = max{m

j₁

, . . . , m

j_P

} and consider D

j₁

, . . . , D

j_P

as defined previously. Since

j_P

X

k=1

σ

_ε^k

2

≤ C

j_P

X

k=1

σ

^k_ε

2

and u

_ε

≤ u

^j_ε

for any j, we have the following inequality

F

ε,a

(σ

ε

, u

ε

; supp(σ

^j_ε¹

) ∩ · · · ∩ supp(σ

_ε^jP

)) ≤ C

jP

X

k=j1

F

ε,a

(σ

^k_ε

, u

^k_ε

; D

k

)

And by inequality (3.15) follows

F

ε,a

(σ

ε

, u

ε

; supp(σ

_ε^j1

) ∩ · · · ∩ supp(σ

_ε^jP

)) ≤ C





(1 − η)

²

ε

ⁿ⁻¹

jP

X

k=j₁

L

ⁿ

(D

k

) + 2 m

²

r

²_∗

η ε

ⁿ⁻²



 .

Which vanishes as ε ↓ 0. Let us remark that the intersection supp(σ

_ε^j1

) ∩ supp(σ

_ε^j2

) ∩ · · · ∩ supp σ

_ε^jP

is non empty

for any ε only if the segments Σ

j₁

, . . . , Σ

j_P

have a common point. Since we are considering a polyhedral vector

measure composed by N segments the worst case scenario is that we have 2N intersections in which at most N