DAMAGE-DRIVEN FRACTURE WITH LOW-ORDER POTENTIALS: ASYMPTOTIC BEHAVIOR, EXISTENCE AND APPLICATIONS

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DAMAGE-DRIVEN FRACTURE WITH LOW-ORDER POTENTIALS: ASYMPTOTIC BEHAVIOR, EXISTENCE AND APPLICATIONS

Marco Caroccia, Nicolas van Goethem

To cite this version:

Marco Caroccia, Nicolas van Goethem. DAMAGE-DRIVEN FRACTURE WITH LOW-ORDER POTENTIALS: ASYMPTOTIC BEHAVIOR, EXISTENCE AND APPLICATIONS. 2018. �hal- 01963478�

DAMAGE-DRIVEN FRACTURE WITH LOW-ORDER POTENTIALS:

ASYMPTOTIC BEHAVIOR, EXISTENCE AND APPLICATIONS

MARCO CAROCCIA AND NICOLAS VAN GOETHEM

Contents

1. Introduction 1

2. Notations and preliminaries 3

2.1. Settings of the problem 4

2.2. Main Theorems 5

3. Liminf inequality 6

4. Limsup inequality 15

4.1. Recovery sequence in Cl(Ω;Rⁿ) 16

4.2. Recovery sequence foru∈SBD²(Ω) 24

5. Compactness result and minimum problem 27

5.1. Compactness 27

5.2. Statement of the minimum problem 28

5.3. Recovery sequence with prescribed boundary condition 29

5.4. Proof of Theorem 5.2 34

6. Selected applications 35

6.1. A simple model of fracking 36

6.2. Pressure almost constant inx: the two-rocks model 37

6.3. A model of plastic slip: F =p|e(u)| 38

6.4. The Tresca yield model in elasto-plasticity: F =λ_max(Ae(u))−λ_min(Ae(u)) 39

6.5. The non-interpenetration condition 39

7. Appendix 40

7.1. A semicontuity result onSBD 40

Acknowledgements 43

References 44

Abstract. We study the Γ-convergence of damage to fracture energy functionals in the presence of low-order nonlinear potentials that allows us to model physical phenomena such as fluid-driven fracturing, plastic slip, and the satisfaction of kinematical constraints such as crack non-interpenetration. Existence results are also addressed.

1. Introduction

In linearized elasticity, the simplest model of damage-driven brittle fracture assumes that a scalar 0 ≤ v ≤ 1 multiplies the elasticity tensor, that is thus weakened in the damage region. At the same time, following Griffith-Bourdin-Francfort-Marigo approach [29,30,9], a certain amount of energy is dissipated in the damage region, and one seeks the minimum of the total energy consisting of the sum of the elastic stored energy and the dissipation terms. Specifically, in [1] the following damage-dependent energy functional

Key words and phrases. special function of bounded deformation, fracture, free discontinuity problem, Γ-convergence, phase-field approximation, geometric measure theory.

was considered¹:

Jε(u, v) :=

Z

Ω

vAe(u)·e(u)dx+1 ε

Z

Ω

ψ(v)dx, (1.1)

with ψ(v) = k in the damage region ω ⊂ Ω, zero elsewhere, k a material-dependent damage coefficient,v≥αεwithα >0, and whereεrepresent the thickness of the damaged region, also related to the mesh size. Here A stands for one half the constant isotropic elasticity tensor. The numerical simulations done in [1] have shown that model consistency under mesh refinement strongly depended on the ratio k/ε. Indeed Eq. (1.1) was used for numerical purposes as a phase-field approximation of Griffith-like energies for crack, though, without studying any rigorous convergence result asε→0. The aim of this work is to study this convergence for a generalized model including low-order potentials.

The so-called Griffith energy reads JG(u) :=

Z

Ω

Ae(u)·e(u) dx+kHⁿ⁻¹(Ju). (1.2) In anti-plane elasticity, though, that is, withAone half the identity tensor, e(u) replaced by ∇u where u is the vertical component of the displacement field, it is well-known that (1.2) is approximated in the sense of Γ-convergence by the Ambrosio-Tortorelli functional

ATε(u, v) :=

Z

Ω

(v+ηε)|∇u|²+(1−v)²

ε +ε|∇v|²

dx, (1.3)

where it is crucial for the residual damage to be of orderη_ε =o(ε). A general case study in function of this parameterηε with Γ-convergence results in the anti-plane case was carried out in [32] as based on Ambrosio-Tortorelli approximation, whereas an approximation of the type (1.1) had been considered for the scalar case, slightly earlier by the same authors in [24]. In real elasticity, that is, for the vectorial u and its symmetric gradient e(u) (as well as inn-dimensions), the first significant Γ-convergence convergence result is found in [27], with an Ambrosio-Tortorelli-like approximation. Recently, existence results for the original Griffith’s functional have been provided in 2D passing by Korn-type inequalities in GSBD space [31,19] (see also [16] and [15]). In [17,18] the authors manage to get rid of any artificial integrability condition on the displacement field by carefully approximating the singularities, and prove some existence results by Γ-convergence with the topology of measures.

In the present paper, with the topology ofL¹, we are concerned with approximations as based on functionals of the type (1.1). Indeed, it is closer to the numerical method chosen for simulation of damage-driven fracture, in particular as far as topological sensitivity analysis is performed, already in [1] and more recently in [36]. In particular, we stick to a simple first-order damage energy, i.e., without gradients of v in the energy functional (see [7] for other gradient-free approximations in other contexts). Note however, that the gradient constraint is found in the admissibility class, which from a technical viewpoint has the same effect. The first aim of this work was to justify from a mathematical perspective a simple model of fracking based on damage and fluid-driven fracture and the topological derivative concept [37]. In that work, numerical simulations were performed, based on the minimization of an energy functional of the type

F_ε(u, v) :=J_ε(u, v)−p Z

Ω

ψ(v)divudx, (1.4)

that models a crack filled with a fluid with an imposed hydrostatic pressure p which is quasi-statically increased in order to trigger a crack opening. As a generalization of this

1Here we discard on purpose the work of the external forces.

problem, our main goal in this paper is to study the asymptotic behaviour, inε, of general functionals with low-order potential of the form

F_ε(u, v) :=J_ε(u, v) + Z

Ω

F(x, e(u), v)dx, (1.5)

whereF need not to be positive. In particular, fracking is recovered forF =−pψ(v)trace(e(u)), but it happens that other interesting cases can be studied as for instance (i) hydraulic frac- ture in porous media, (ii) plastic slip, (iii) non-interpenetration or Tresca-type conditions, just to cite some applications that we have chosen. Our main result is the Γ-convergence ofF_ε(u, v) to the limitcohesive functional

Φ(u) :=

Z

Ω

Ae(u)·e(u) dx+bHⁿ⁻¹(J_u) +a

Z

Ju

pA([u](z)ν(z))·([u](z)ν(z)) dHⁿ⁻¹(z) +

Z

Ω

F(x, u,1) + Z

Ju

F∞(z,[u]ν) dHⁿ⁻¹(z),

for some appropriate coefficients a and b related to the choice of the damage potential ψ and with F∞ denoting the recession function of the convex potential, i.e., coding the asymptotic behaviour of F as |e(u)| → ∞. Compactness and an original approach to existence results are also proposed in Section 5 (to be precise, the so-called existence of weak type), as well as some general results given in the Appendix. Let us remark that a specific such low-order potential together with a treatment of the Dirichlet boundary condition were also considered in the anti-plane case in [5], with the additional condition that F ≥0, a restriction that we wanted to avoid in the present work. Let us emphasize that in this work, in contrast with the aforementioned results, we address and solve the complete problem that consists in avoiding anyL^∞-bound on the displacement field.

Moreover, our aim is also to be entirely self-contained, in order for these computations and techniques be available for the mathematical/mechanical communities in the clearest way possible. Therefore, some known results are recalled and proven in our Appendix.

Precise bibliography is always provided when cross-references applies, while otherwise our arguments and proof strategy are originals. Specific references for this topic are [24] and [32] while general and fundamental results are found in [35,10,6,28,23,2,3,8,14].

2. Notations and preliminaries

We denote by M_sym^n×n the set of all symmetric matrices with real coefficient. Given an open bounded set Ω with Lipschitz boundary we say that a function u ∈ L¹(Ω;Rⁿ) is a function of bounded deformation if there exists a matrix-valued Radon measure ((Eu)_ij)ⁿ_i,j=1 such that for alli, j= 1, . . . , n it holds

h(Eu)_i,j, ϕi=−1 2

Z

Ω

ui

∂ϕ

∂xj

+uj

∂ϕ

∂xi

dx.

for all ϕ∈C_c^∞(Ω;Rⁿ).

Notice that, if u_k ∈ BD(Ω) and u_k → u in L¹, then Eu_k *^∗ Eu. The space of such functions is endowed with the norm

kuk_BD:=kuk_L1(Ω)+|Eu|(Ω)

where, for any given Radon measureµ,|µ|stands for its total variation. For any sequence {u_k}_k∈N bounded in this norm, up to a subsequence, it holds u_k→ u inL¹. Analogously to the behavior of the function of Bounded Variation we can identify three distinct part of the matrix valued measureEu: the absolutely continuous part, the jump part (supported

on J_u, an (n−1)-rectifiable set) and a Cantor part. Namely, for a generic u∈BD(Ω), we can write

Eu=e(u)Lⁿ+ [u]νuHⁿ⁻¹xJu+E^cu

whereν_u(x) is any unitary vector field orthogonal toJ_u, [u] =u⁺−u⁻the jump ofuwith u^± the approximate limit ofu as we approach Ju and

[u]ν_u := [u]⊗ν_u+ν_u⊗[u]

2 .

Note that in general symbolstands for the symmetric sum. Finally we define the space SBD²(Ω;Rⁿ) as follows:

SBD²(Ω;Rⁿ) :={u∈BD(Ω;Rⁿ) |E^cu= 0, e(u)∈L²(Ω;M_sym^n×n), Hⁿ⁻¹(J_u)<+∞}.

2.1. Settings of the problem. We consider a fourth order tensor A: M_sym^n×n → M_sym^n×n such that there exist a constant κ for which

κ⁻¹|M|²≤AM·M ≤κ|M|²

where M ·L := tr(M L^T) is the standard scalar product inducing the Frobenius norm which, for a generic M ∈M_sym^n×n, is here denoted by |M|.

Having fixed α >0 we define

V_ε:={v∈W^1,∞(Ω)|εα < v≤1, |∇v| ≤1/ε}.

With these notation we define the sequence of energy functionalsF_ε:L¹(Ω;Rⁿ)×L¹(Ω; [0,1])→ R+ to be

F_ε(u, v) :=













 Z

Ω

vAe(u)·e(u)+ 1 ε

Z

Ω

ψ(v)dx +

Z

Ω

F(x, e(u), v) dx

if (u, v)∈H¹(Ω;Rⁿ)×V_ε

+∞ otherwise

(2.1) where ψ is any strictly decreasing (and thus we will often use that ψ(0) > 0), convex function such thatψ(1) = 0 andFis a generic potential subject to the following hypothesis.

Assumption 2.1(On the potentialF). The functionF :Rⁿ×M_sym^n×n×[0,1]→Rsatisfies the following properties:

1) F(·, M,0)is Lipschitz continuous uniformly in M ∈M_sym^n×n; 2) F(x,·,0) and F(x,·,1) are convex for all x∈Ω;

3) −σ|M| ≤F(x, M, v) ≤`|M|, for all (x, M, v) ∈ R<sup>n</sup>×M<sub>sym</sub><sup>n×n</sup>×[0,1] where ` > 0 can be any real constant and

0< σ < max

λ∈(0,1)

( 2p αψ(λ)

√κ(1 + 2p

α|Ω|ψ(λ)/λ) )

<2

rαψ(0)

κ ; (2.2)

4) having set

ω_F(s; 1) := sup

|F(x, M, s)−F(x, M,1)|

|M| : (x, M)∈Rⁿ×M_sym^n×n

, ω_F(s; 0) := sup

|F(x, M, s)−F(x, M,0)|

|M| : (x, M)∈Rⁿ×M_sym^n×n

, then

s→1limω_F(s; 1) = lim

s→0ω_F(s; 0) = 0.

Remark 2.2. In particular, F can be taken as negative as we want by simply taking αψ(0) large enough.

Remark 2.3. We remark that, for any fixed x, since f(M) = F(x, M,0) is convex and satisfiesf(M)≤`|M|, then f is a Lipschitz function with constant `. Indeed, consider a convex functionf :Rⁿ → R (with n >1) such that f(x) ≤`|x|and notice that, for any v ∈ S<sup>n−1</sup>, g(t) := f(x+tv) is still convex and meets the requirement g(t) ≤ `|x+tv|.

In particular limt→+∞g(t)−g(0)

t ≤ ` and since the map t 7→ ^g(t)−g(0)_t is increasing we get

g(t)−g(0)

t ≤`for all t∈R, leading tog<sup>0</sup>(0)≤`and thus to

∇f(x)·v≤` for all v∈S<sup>n−1</sup> ⇒ |∇f(x)| ≤`.

We are interested in the asymptotic behavior (as ε → 0) of the sequence of energies (2.1). In particular the first aim of this paper is to show that the family of functional F_ε, under the assumptions in2.1, is Γ-converging to the energy

Φ(u) :=

Z

Ω

Ae(u)·e(u) dx+ Z

Ω

F(x, u,1) dx +a

Z

Ju

p

A([u](z)ν(z))·([u](z)ν(z)) dHⁿ⁻¹(z) +bHⁿ⁻¹(J_u) +

Z

Ju

F∞(z,[u]ν) dHⁿ⁻¹(z)

defined foru ∈ SBD²(Ω;Rⁿ) and extended to +∞ otherwise. Here we have set, for the sake of shortness,

a= 2p

αψ(0), b= 2

Z 1 0

ψ(t) dt and

F∞(z, M) := lim

t→+∞

F(z, tM,0)−F(z,0,0)

t forz∈Ju and M ∈M_sym^n×n

(see Proposition7.1 to see whyF∞ is well defined for potential F satisfying assumptions 2.1).

Remark 2.4. Notice that the role of the conditionα >0 is linked, at least in the present analysis, to the possibility for F to be negative. The approach here proposed seems to work also if we replace the condition vε ≥αε with the condition vε ≥ ηε for an ηε such thatη_ε/ε→0, provided F ≥0.

2.2. Main Theorems. Setting, F :L¹(Ω;Rⁿ)×L¹(Ω; [0,1])→R+ to be

F(u, v) :=











Φ(u), ifu∈SBD²(Ω)

and v= 1Lⁿ-a.e. in Ω, +∞ otherwise

(2.3)

we are able to provide the following Γ-convergence result:

Theorem 2.5. Provided the notations and the assumptions introduced in Subsection 2.1 we have

Γ- lim

ε→0 F_ε=F

on the spaceH¹(Ω;Rⁿ)×Vε⊂SBD²(Ω)×L¹(Ω)with respect to the convergence induced by theL¹ topology. In particular, the following assertions hold true:

a) For any (u_ε, v_ε)∈H¹(Ω;Rⁿ)×V_ε such that u_ε →u, v_ε→v in L¹ we have lim inf

ε→0 F_ε(uε, vε)≥ F(u, v);

b) Let {ε_j}_j∈N be a vanishing sequence. Then for any u ∈ SBD²(Ω) there exists a subsequence{ε_jk}_k∈N⊂ {ε_j}_j∈N and (uk, vk)∈H¹(Ω;Rⁿ)×Vε_jk such that

u_k→u, v_k→1 in L², and lim

k→+∞F_εjk(u_k, v_k) =F(u,1).

Let us remark that, assertion b) allows us to recover the energy of any u ∈SBD²(Ω), which consist of an important improvement of the results in [27], where onlyu∈SBD²(Ω)∩

L^∞(Ω;Rⁿ) can be recovered. This improvement is mostly due to the recent refinement [22] of the approximation theorem for GSBD function contained in [32]. Such a theorem yields a more precise information about the lack of energy on the jump set between the function u and its (more regular) approximants (see Property d) of Theorem4.7 below).

Thank to this recent result, our solution is sharp, since the complete problem is addressed, i.e., without theL^∞-bound, as found in most results about this problem.

Moreover, we prove that the sequences with bounded energy are compact with respect to theL¹ topology. Namely the following theorem holds true:

Theorem 2.6. With the notations and the assumptions introduced in Subsection 2.1, if (uε, vε)∈H¹(Ω;Rⁿ)×Vε are sequences such that

sup

ε

{ku_εk_L1 +F_ε(uε, vε)}<+∞ (2.4) there exists two subsequences {(u_εk, v_εk)}_k∈N⊂ {(u_ε, v_ε)}_ε>0 and u∈SBD²(Ω)such that

uεk →u, vεk →1 in L¹ and Euε_k *^∗Eu. Moreover, for any λ∈(0,1) it holds

e(u_k)1{v_εk≥λ} * e(u) in L²(Ω;M_sym^n×n).

The proof of Theorem2.5is obtained by separately proving statement a) (in Section3, Theorem3.1) and statement b) (in Section4, Theorem4.10). The compactness Theorem is proven in Subsection5.1and it is basically a consequence of Propositions3.2and 3.3in Section 3. For the existence of minimizers with prescribed Dirichlet boundary condition we send the reader to Subsection 5.2 where, under specific additional hypothesis on the potential F, on the boundary data and on the domain, the relaxed problem over Ω is treated. We finally provide some examples of applications in Section 6.

3. Liminf inequality

This section is entirely devoted to the proof of the following theorem:

Theorem 3.1. Given (uε, vε) ∈H¹(Ω;Rⁿ)×Vε such that uε →u in L¹ and vε→ v a.e.

it holds

lim inf

ε→0 F_ε(uε, vε)≥ F(u, v).

To achieve the proof we will analyze separately what happens on the energy restricted on the sequence of sets Ω^λ_ε ={v_ε ≥λ}and Ω\Ω^λ_ε. We start by first gaining some information on the sequences with bounded energy. To do that we will exploit the hypothesis on the nonlinear potential F. Let us denote by

W_ε(u, v) :=









 Z

Ω

vAe(u)·e(u) +ψ(v) ε

dx if (u, v)∈H¹(Ω;Rⁿ)×Vε

+∞ otherwise

(3.1)

and let us observe that

F_ε(u, v) =W_ε(u, v) + Z

Ω

F(x, e(u), v) dx.

We underline that any bounds of the type sup

ε>0

{W_ε(uε, vε)}<+∞

leads, as we will discuss below, to an information on the convergence ofuε, vε. We now show how to derive such kind of control starting from the boundedness ofF_ε. In the sequel we adopt the notationsW_ε(u, v;A),F_ε(u, v;A),F(u, v;A) meaning the usual energies localized to the set A.

Proposition 3.2. Under the hypothesis stated in Subsection 2.1 on A, ψ and F, there exists a constant C depending onα,A,|Ω|, ψ andσ only such that

W_ε(u, v;A)< C(F_ε(u, v;A) + 1) (3.2) for all(u, v)∈H¹(Ω;Rⁿ)×V_ε and for all open sets A⊆Ω.

Proof. The key point is the estimate Z

A

F(x, e(u), v) dx≥ −σ Z

A

|e(u)|dx. (3.3) Set

Ω^λ ={v≤λ}

and notice that Z

A

|e(u)|dx= Z

A\Ω^λ

|e(u)|dx+ Z

A∩Ω^λ

|e(u)|dx and that

Z

A\Ω^λ

|e(u)|dx≤p

|A|

Z

A\Ω^λ

|e(u)|²dx

!1/2

≤ r|A|

λ Z

A\Ω^λ

v|e(u)|²dx

!1/2

≤√ κ

r|A|

λ

pW_ε(u, v;A). (3.4) On the other hand,

Z

A∩Ω^λ

|e(u)|dx=

√κ 2p

αψ(λ) Z

A∩Ω^λ

2p

κ⁻¹ψ(λ) rαε

ε |e(u)|dx

≤

√κ 2p

αψ(λ) Z

A∩Ω^λ

αεκ⁻¹|e(u)|²dx+ Z

A∩Ω^λ

ψ(λ) ε dx

≤

√κ 2p

αψ(λ) Z

A∩Ω^λ

vAe(u)·e(u) dx+ Z

A∩Ω^λ

ψ(v) ε dx

≤

√κ 2p

αψ(λ)W_ε(u, v;A). (3.5)

In particular, by combining (3.3),(3.4) and (3.5) we obtain, for any (u, v)∈H¹(Ω;Rⁿ)×V_ε Z

A

F(x, e(u), v) dx≥ −σ rκ

α

"r α|Ω|

λ

pW_ε(u, v;A) + 1 2p

ψ(λ)W_ε(u, v;A)

#

≥ −σ rκ

α(1 +W_ε(u, v;A))

"r α|Ω|

λ + 1

2p ψ(λ)

#

=−σ(1 +W_ε(u, v;A))

"√

κ(1 + 2p

α|Ω|ψ(λ)/λ) 2p

αψ(λ)

#

, (3.6)

where we have used the fact that p

W_ε(u, v;A) and W_ε(u, v;A) are each always bounded by (1 +W_ε(u, v;A)). Moreover, inequality (3.6) holds for anyλ∈(0,1) and hence it holds for the minimum amongλwhich means that

Z

A

F(x, e(u), v) dx≥ −σ(1 +W_ε(u, v;A)) min

λ∈(0,1)

(√

κ(1 + 2p

α|Ω|ψ(λ)/λ) 2p

αψ(λ)

) . Notice that Assumption 1) in 2.1requires that

σ < max

λ∈(0,1)

( 2p αψ(λ)

√κ(1 + 2p

α|Ω|ψ(λ)/λ) )

= min

λ∈(0,1)

(√

κ(1 + 2p

α|Ω|ψ(λ)/λ) 2p

αψ(λ)

)!−1

. In particular for someδ >0 depending onα,A,|Ω|, ψ and σ only we have

σ min

λ∈(0,1)

(√

κ(1 + 2p

α|Ω|ψ(λ)/λ) 2p

αψ(λ)

)

≤(1−δ) leading to

Z

A

F(x, e(u), v) dx≥ −(1−δ)(1 +W_ε(u, v;A)). (3.7) By exploiting (3.7) we reach

F_ε(u, v;A) =W_ε(u, v;A) + Z

A

F(x, e(u), v) dx

≥ W_ε(u, v;A)−(1−δ)W_ε(u, v;A)−(1−δ)≥δW_ε(u, v;A)−1

which, by setting C =δ⁻¹, achieves the proof.

Let us now analyze the behaviour of the part of the energy that lives on the set{v_ε ≥λ}.

We set up some notation that will be repeatedly used in this subsection. Given a sequence {v_ε}_ε>0 ⊆Vε and a fixedλ∈(0,1) we define

Ω^λ_ε ={v_ε≤λ}.

We also set

I_ε¹(λ) :=

Z

Ω\Ω^λ_ε

v_εAe(u_ε)·e(u_ε) dx, I_ε²(λ) :=

Z

Ω\Ω^λ_ε

ψ(vε) ε dx, I_ε³(λ) :=

Z

Ω^λ_ε

vεAe(uε)·e(uε) +ψ(vε) ε

dx.

Then

F_ε(u_ε, v_ε; Ω^λ_ε) =I_ε³(λ) + Z

Ω^λ_ε

F(x, e(u_ε), v_ε)dx

is the part of the energy that will provide the jump terms in the limit, as Proposition 3.4 will show. Let us first treat the bulk part F_ε(uε, vε)− F_ε(uε, vε; Ω^λ_ε) = I_ε¹(λ) +I_ε²(λ) + R

Ω\Ω^λ_ε F(x, e(u_ε), v_ε).

Proposition 3.3. Let (u_ε, v_ε)∈H¹(Ω;Rⁿ)×V_ε be such that uε →u, vε→v in L¹ and with

sup

ε>0

{F_ε(u_ε, v_ε)}<+∞. (3.8)

Then

sup

ε>0

Z

Ω

|e(u_ε)|dx

<+∞. (3.9)

Moreover u∈SBD²(Ω;Rⁿ),v= 1 a.e. inΩ and for any λ >0 it holds sup

ε>0

(Z

Ω\Ω^λ_ε

|e(u_ε)|²dx, )

<+∞, (3.10)

lim inf

ε→0

Z

Ω\Ω^λ_ε

v_εAe(u_ε)·e(u_ε) +ψ(v_ε)

ε +F(x, e(u_ε), v_ε)

dx

≥ Z

Ω

[Ae(u)·e(u) +F(x, e(u),1))] dx+ 2(h(1)−h(λ))Hⁿ⁻¹(Ju)

(3.11) where

h(t) :=

Z t 0

ψ(τ) dτ.

Proof. Thanks to Proposition 3.2, the bound (3.8) implies sup

ε>0

{W_ε(uε, vε)}<+∞. (3.12) In particular

Z

Ω

ψ(vε) dx→0

which implies ψ(v) = 0 a.e. in Ω and thus v = 1 a.e. in Ω. Moreover, fix λ∈(0,1) and notice that

I_ε¹(λ) = Z

Ω\Ω^λ_ε

vAe(u)·e(u) dx≥λκ⁻¹ Z

Ω\Ω^λ_ε

|e(u)|²dx (3.13) and

I_ε³(λ) = Z

Ω^λ_ε

vAe(u)·e(u) +ψ(v_ε) ε

dx≥

Z

Ω^λ_ε

κ⁻¹αε|e(u)|²+ψ(v_ε) ε

dx

≥2√ α

√ κ⁻¹

Z

Ω^λ_ε

|e(u)|p

ψ(vε) dx≥p

κ⁻¹αψ(λ) Z

Ω^λ_ε

|e(u)|dx. (3.14) Inequality (3.13) implies (3.10), while (3.14), after a further application of Cauchy-Schwarz inequality in (3.13), yields (3.9), that in turn establishes the weak compactness in BD.

Such a compactness in the weak topology of BD, together with uε → u in L¹, implies u ∈ BD(Ω). The remaining part of the proof is obtained as a slight variation of the original arguments of [27] extended in such a way as to take into account the nonlinear potential part.

Step one: proof that u∈SBD²(Ω). We start from the fact that sup

ε>0

{I_ε¹(λ) +I_ε²(λ) +I_ε³(λ)}= sup

ε>0

{W_ε(u_ε, v_ε)}<+∞ for everyλ∈(0,1)

which implies a uniform bound inεon eachI_εⁱfori= 1,2,3. Thanks to the co-area formula and to the property of vε ∈ Vε (in particular to|∇v_ε|<1/ε) we obtain, by recalling the definition of h

I_ε²(λ) = Z

Ω\Ω^λ_ε

ψ(vε) ε dx≥

Z

Ω\Ω^λ_ε

|∇v_ε|ψ(v_ε) dx= Z

Ω\Ω^λ_ε

|∇h(v_ε)|dx

= Z h(1)

h(λ)

P({h(v_ε)> t}; Ω) dt≥(h(1)−h(λ))P({h(v_ε)> tε}; Ω),

where in the last inequality we considered the mean value theorem to findtε∈(h(λ), h(1)).

We now set

λε:=h⁻¹(tε)∈(λ,1) and observe that

P(Ω\Ω^λ_ε^ε; Ω)≤CI_ε²(λ), yielding

sup

ε>0

{P(Ω\Ω^λ_ε^ε; Ω)}<+∞.

Consideruε:=uε1_Ω\Ω^λε_ε and notice that, since vε →1 (and thus|Ω\Ω^λ_ε^ε| → |Ω|), we have u_ε→u inL¹. It is easy to see that, as a consequence of the chain rule [4, Theorem 3.96]

(see [4, Example 3.97],uε is aBV function with Du_ε=1_Ω\Ω^λε_ε ∇u_εLⁿ+u_ε⊗ν_Ωλε

ε Hⁿ⁻¹x∂^∗Ω^λ_ε^ε.

Moreover the above formula implies thatHⁿ⁻¹(Juε\∂^∗Ω_ε^λε) = 0. Thusuε∈SBD(Ω;Rⁿ)∩ L²(Ω) (since also|u_ε| ≤ |u_ε| ∈L²). In particular

sup

ε>0

{Hⁿ⁻¹(J_uε)}<+∞.

From (3.13) we also get thatu_ε∈SBD²(Ω;Rⁿ)∩L²(Ω) with sup

ε>0

Z

Ω

|e(u_ε)|²dx+Hⁿ⁻¹(J_uε)

<+∞. (3.15)

By applying [14, Lemma 5.1] this gives us thatu∈SBD²(Ω;Rⁿ), since

e(u_ε)* e(u) weakly inL²(Ω;M_sim^n×n). (3.16) Hⁿ⁻¹(Ju)≤lim inf

ε→0 Hⁿ⁻¹(Juε). (3.17)

Step two: proof of (3.11). Remark that the sequence {λ_ε}_ε>0 defined above lies in the interval (λ,1). In particular Ω\Ω^λ_ε^ε ⊆ Ω\Ω^λ_ε and relation (3.16), due to the convexity of the map M 7→ AM ·M and to the strong convergence of v_ε to 1 almost everywhere, means that (see for instance [10, Theorem 2.3.1])

lim inf

ε→0

Z

Ω\Ω^λ_ε

v_εAe(u_ε)·e(u_ε) dx≥lim inf

ε→0

Z

Ω\Ω^λεε

v_εAe(u_ε)·e(u_ε) dx

= lim inf

ε→0

Z

Ω

vεAe(uε)·e(uε) dx

≥ Z

Ω

Ae(u)·e(u) dx. (3.18)

Moreover Z

Ω\Ω^λ_ε

F(x, e(uε), vε) dx− Z

Ω\Ω^λ_ε

F(x, e(uε),1) dx

≤ Z

Ω\Ω^λ_ε

ωF(vε; 1)|e(u_ε)|dx

Z

Ω\Ω^λ_ε

F(x, e(u_ε),1) dx− Z

Ω\Ω^λε_ε

F(x, e(u_ε),1) dx

≤ Z

Ω^λεε \Ω^λ_ε

`|e(u_ε)|dx,

where we exploited item 3): |F(x, M, v)| ≤`|M|of Assumption 2.1. The above quantities are vanishing (by item 4) of Assumption2.1 onF, thanks to the fact that |Ω^λ_ε^ε\Ω^λ_ε| →0 and thanks to (3.10)) and hence this fact, together with the convexity of the map M → F(x, M,1), implies (using once again (3.16) and the semicontinuity Theorem [10, Theorem 2.3.1])

lim inf

ε→0

Z

Ω\Ω^λ_ε

F(x, e(u_ε), v_ε) dx= lim inf

ε→0

Z

Ω\Ω^λεε

F(x, e(u_ε),1) dx

≥ Z

Ω

F(x, e(u),1) dx. (3.19)

To achieve the proof of (3.11) we need only to show that lim inf

ε→0

Z

Ω\Ω^λ_ε

ψ(vε)

ε dx≥2(h(1)−h(λ))Hⁿ⁻¹(Ju).

In particular we use the fact that lim inf

ε→0 P({h(v_ε)≥t}; Ω)≥2Hⁿ⁻¹(J_u) for all t∈(h(λ), h(1)) (3.20) proved in [27] via a slicing argument as established also in [26, 25, Lemma 3.2.1]. Relation (3.20) implies immediately that

Z

Ω\Ω^λ_ε

ψ(vε) ε dx≥

Z h(1) h(λ)

P({h(v_ε)≥t}; Ω) dt≥2(h(1)−h(λ))Hⁿ⁻¹(Ju) leading to

lim inf

ε→0

Z

Ω\Ω^λ_ε

ψ(vε)

ε dx≥2(h(1)−h(λ))Hⁿ⁻¹(Ju). (3.21) By collecting (3.18), (3.19) and (3.21) we deduce (3.11).

We now provide the liminf inequality for the (asymptotically equivalent) remaining part of the energy on Ω\Ω^λ_ε. In order to do so, we will need to apply Proposition 7.7, stated in the Appendix, that is a well-known approach (inspired by [11]) when dealing with local functionals. We will also use the blow-up technique originally designed in [28].

Proposition 3.4. Let (uε, vε)∈H¹(Ω;Rⁿ)×Vε be such that uε →u, vε→v in L¹ and with

sup

ε>0

{F_ε(u_ε, v_ε)}<+∞. (3.22) Suppose also that, at a given z∈Ju such that

r→0lim

|Eu|(B_r(z))

ωn−1rⁿ⁻¹ =|[u](z)ν(z)| (3.23) we have

r→0limlim inf F_ε(u_ε, v_ε;B_r(z))

rⁿ⁻¹ <+∞,

then, for every λ∈(0,1), it holds

r→0limlim inf

ε→0

1 rⁿ⁻¹

Z

Br(z)∩Ω^λ_ε

[2p

αψ(0)p

Ae(uε)·e(uε) +F(x, e(uε),0)] dx

≥2p

αψ(0)p

A[u](z)ν(z)·[u](z)ν(z) +F∞(z,[u](z)ν(z),0).

(3.24)

Proof. SetG:Rⁿ×M_sym^n×n→R⁺ to be G(x, M) = 2√

αp ψ(0)√

AM·M +F(x, M,0)

and notice, by the hypothesis on F, that G(x,·) is a positive convex functions on M_sym^n×n for any x∈Ω. In particularG satisfies the hypothesis of Proposition7.7 and thus

lim inf

ε→0

Z

Br(z)

G(x, e(u_ε)) dx≥ Z

Br(z)

G(x, e(u)) dx+ Z

Ju∩Br(z)

G∞(y,[u]ν) dHⁿ⁻¹(y) (3.25) for any B_r(z)⊂Ω. Let ε_k be the sequence achieving

lim inf

ε→0

Z

Ω^λ_ε

G(x, e(uε)) dx= lim

k→+∞

Z

Ω^λ_εk

G(x, e(uεk)) dx and define, for every measurable setA⊆Ω, the Radon measures:

µ_k(A) :=

Z

A∩Ω^λ_εk

G(x, e(uε_k)) dx, ξ_k(A) :=

Z

A

G(x, e(u_εk)) dx,

Notice that, due to the uniform bound on the energy F_ε we have sup

ε

{µ_k(Ω)}<+∞, sup

ε

{ξ_k(Ω)}<+∞,

and thus, up to a subsequence (not relabeled), we can find Radon measuresµ, ξ such that µ_k*^∗ µ, ξ_k*^∗ ξ.

Step one: We assert that the proof of (3.24) follows easily from the following fact:

r→0lim

µ(B_r(z)) rⁿ⁻¹ = lim

r→0

ξ(B_r(z))

rⁿ⁻¹ . (3.26)

Indeed, by assuming the validity of (3.26) we conclude that, for L¹−a.e. r >0 it holds (because of (3.25))

ξ(Br(z)) rⁿ⁻¹ = 1

rⁿ⁻¹ lim

k→+∞ξ_k(Br(z))≥ 1 rⁿ⁻¹

Z

Ju∩Br(z)

G∞(y,[u]ν) dHⁿ⁻¹(y), implying

r→0lim

µ(B_r(z)) rⁿ⁻¹ = lim

r→0

ξ(B_r(z))

rⁿ⁻¹ ≥G∞(z,[u](z)ν(z)) forHⁿ⁻¹-a.e. z∈Ju. This gives

µ(A)≥ Z

Ju∩A

G∞(y,[u]ν) dHⁿ⁻¹(y) and since

G∞(y, M) = 2p

αψ(0)√

AM·M+F∞(y, M)

we obtain, forA= Ω, relation (3.24).

Step two: Let us focus on (3.26). It suffices to check that

r→0limlim inf

k→+∞

1 rⁿ⁻¹

Z

Br(z)∩(Ω\Ω^λ_εk)

G(x, e(uε_k)) dx= 0.

Setτ = max{σ, `}. We claim that

r→0limlim inf

k→+∞

W_εk(u_εk, v_εk;B_r(z))

rⁿ⁻¹ <+∞.

Indeed, by defining for a generic Borel setA⊆Ω, the measures ζ_k(A) :=W_εk(uε_k, vε_k;A)

and by exploiting the uniform bounds onF_ε and Proposition3.2 we have, up to a subse- quence,ζk*^∗ ζ. Moreover, due to (3.5)

F_εk(u_εk, v_εk;A) =ζ_k(A) + Z

A

F(x, e(u_εk), v_εk) dx.

and

Z

A

F(x, e(uε_k, vε_k) dx≥ −σ

"

Z

A\Ω^λ_εk

|e(u_εk)|dx+ Z

A∩Ω^λ_εk

|e(u_εk)|dx

#

and by repeating the computation in (3.4), (3.5) we achieve Z

A\Ω^λ_εk

|e(u_εk)|dx≤ r

κ|A|ζ_k(A)

λ ,

Z

A∩Ω^λ_εk

|e(u_εk)|dx≤ r κ

αψ(λ)ζ_k(A).

In particular

k→+∞lim 1 rⁿ⁻¹

p|B_r(z)|ζ(B_r(z)) =√ ω_n√

r

rζ(B_r(z)) rⁿ⁻¹

and this yields, still following the computation giving (3.6), (3.7), for r small enough and for someδ >0 depending onα,A,|Ω|, ψ and σ only

lim inf

k→+∞

1 rⁿ⁻¹

Z

Br(z)

F(x, e(u_εk, v_εk) dx≥ −(1−δ)

ζ(B_r(z)) rⁿ⁻¹ + 1

. Thus

r→0limlim inf

k→+∞

F_εk(uεk, vεk;Br(z)

rⁿ⁻¹ ≥ −1 +C lim

r→+∞

ζ(Br(z)) rⁿ⁻¹ for a positive constantC. Thus, if

r→0limlim inf

ε→0

F_ε(uε, vε;Br(z))

rⁿ⁻¹ <+∞

then

r→0limlim inf

k→+∞

W_εk(u_εk, v_εk;B_r(z))

rⁿ⁻¹ <+∞. (3.27)

Clearly, Z

(Ω\Ω^λ_εk)∩Br(z)

G(x, e(u_εk)) dx

≤ Z

(Ω\Ω^λ_εk)∩Br(z)

(2p

αψ(0)κ+τ)|e(u_εk)|dx

≤(2p

αψ(0)κ+τ) Z

(Ω\Ω^λ_εk)∩B_r(z)

|e(u_εk)|dx

≤C|B_r(z)|^1/2 Z

Ω\Ω^λ_εk∩B_r(z)

|e(u_εk)|²dx

!1/2

.

Thus 1 rⁿ⁻¹

Z

(Ω\Ω^λ_εk)∩B_r(z)

G(x, e(u_εk)) dx

≤Cr^1/2 1 rⁿ⁻¹

Z

(Ω\Ω^λ_ε)∩B_r(z)

|e(u_εk)|²dx

!1/2

≤ Cr^1/2 λ^1/2

1 rⁿ⁻¹

Z

Ω∩Br(z)

v_εk|e(u_εk)|²dx

!1/2

≤ Cr^1/2 λ^1/2

1 rⁿ⁻¹

Z

Ω∩Br(z)

v_εk|e(u_εk)|²+ψ(vε_k) ε_k

dx

!1/2

= Cr^1/2 λ^1/2

1

rⁿ⁻¹W_εk(uεk, vεk;Br(z)) 1/2

. But now, thanks to (3.27), it holds

r→0limlim inf

k→+∞

1 rⁿ⁻¹

Z

(Ω\Ω^λ_εk)∩B_r(z)

G(x, e(u_εk)) dx

= 0,

yielding (3.26), thence completing the proof.

We are now ready to proceed to the proof of Theorem 3.1.

Proof of Theorem 3.1. Let (uε, vε)∈H¹(Ω;Rⁿ)×Vε with uε→u and vε→v inL¹. We can easily assume that sup_ε{F_ε(u_ε, v_ε)}<+∞(otherwise there is nothing to prove).

Let λ∈(0,1) to be chosen later and apply Proposition 3.3 to deduce thatv = 1 Lⁿ-a.e.

in Ω, u∈SBD²(Ω) and to conclude that (3.11) and (3.10) are in force. Thus lim inf

ε→0

Z

Ω\Ω^λ_ε

vεAe(uε)·e(uε) +ψ(vε)

ε +F(x, e(uε), vε)

dx

≥ Z

Ω

[Ae(u)·e(u) +F(x, e(u),1)] dx+ 2(h(1)−h(λ))Hⁿ⁻¹(J_u).

(3.28)

By writing

F_ε(u_ε, v_ε)≥ Z

Ω\Ω^λ_ε

v_εAe(u_ε)·e(u_ε) +ψ(v_ε)

ε +F(x, e(u_ε), v_ε)

dx +

Z

Ω^λ_ε

v_εAe(u_ε)·e(u_ε) +ψ(v_ε)

ε +F(x, e(u_ε), v_ε)

dx,

(3.29)

it is readily seen that it suffices to focus on the second addendum in the right-hand side of (3.29), denoted asG_ε(uε, vε;λ), which by Cauchy-Schwarz inequality yields

G_ε(u_ε, v_ε)≥ Z

Ω^λ_ε

h 2√

αp

Ae(u_ε)·e(u_ε)p

ψ(v_ε) +F(x, e(u_ε), v_ε)i dx.

Since ψ(s) → ψ(0) and ω_F(s; 0) → 0 for s → 0, for some λ_δ we have that |p ψ(s)− pψ(0)|+ωF(s; 0)≤δ for all s < λ_δ. Thus, for a suitably smallλ, we have

Z

Ω^λ_ε

2√ αp

Ae(u_ε)·e(u_ε)(p

ψ(v_ε)−p

ψ(0)) dx

≤2δ√ κα

Z

Ω^λ_ε

|e(u_ε)|dx and

Z

Ω^λ_ε

[F(x, e(uε), vε)−F(x, e(uε),0)] dx

≤ Z

Ω^λ_ε

ωF(vε; 0)|e(u_ε)|dx≤δ Z

Ω^λ_ε

|e(u_ε)|dx.

In particular, according to (3.9), we reach

ε→0lim Z

Ω^λ_ε

[2√ αp

Ae(uε)·e(uε)(p

ψ(vε)−p

ψ(0)) dx + lim

ε→0

Z

Ω^λ_ε

F(x, e(u_ε), v_ε)−F(x, e(u_ε),0)] dx

≤δC

whereC is a constant depending only on the sequence on the sequenceu_ε. In particular, we have

lim inf

ε→0 G_ε(uε, vε)≥ −δC+ lim inf

ε→0

Z

Ω^λ_ε

h 2p

αψ(0)p

Ae(uε)·e(uε) +F(x, e(uε),0) i

dx.

ForHⁿ⁻¹-a.e. z∈Ju we can guarantee that

r→0lim

Eu(B_r(z))

ωn−1rⁿ⁻¹ =|[u](z)ν(z)|

Moreover, if

r→0limlim inf

ε→0

F_ε(uε, vε;Br(z))

rⁿ⁻¹ = +∞

then the (n−1)-dimensional densities of the lim inf lower bound is +∞ and the lim inf inequality trivially holds. Instead, for all the other point it must hold

r→0limlim inf

ε→0

F_ε(uε, vε;Br(z))

rⁿ⁻¹ <+∞.

Hence by applying Proposition3.4, and in particular relation (3.24), we get lim inf

ε→0 G_ε(uε, vε)≥ −δC+ Z

Ju

h 2p

αψ(0)p

A[u]ν·[u]ν+F∞(x,[u]ν,0) i

dx.

(3.30) Summarizing, we have shown that for any δ > 0 there exists a λ_δ such that, if λ ≤ λ_δ, then (3.30) holds true. Moreover (3.28) is in force for everyλ∈(0,1). Thus, for anyδ >0 it must holds

lim inf

ε→0 F_ε(uε, vε)≥ −δC+F(u, v),

that, by taking the limit asδ →0, achieves the proof.

4. Limsup inequality

This section is entirely devoted to the construction of a recovery sequence. We first show how to recover the energy on a special class of function Cl(Ω;Rⁿ) and then we show, with a density argument, that each function u ∈SBD²(Ω;Rⁿ) can be recovered. Let us define

Cl(Ω;Rⁿ) :=



















u∈SBV²(Ω;Rⁿ)∩L^∞(Ω;Rⁿ)∩W^m,∞(Ω\Ju;Rⁿ), for all m∈N whereJu∩Ω is the finite unionS of closed, pairwise disjoint (n−1)-dimensional simplexes intersected with Ω and Hⁿ⁻¹((J_u∩Ω)\J_u) = 0.



















. (4.1)

Figure 4.1. In grey is depicted the set Aϑε. In order to avoid overlaps, since the functionuεhas been defined outside the blue ball of sizeε, we can extend it and sew up everything together inside such a region by exploiting a capacitary argument as briefly sketched in Remarks 4.1 and 4.2. In particular we can always assume that the pieces of the set A_ϑε, on each branches of Ju, do not overlap. In order to alleviate the notations we are neglecting this correction.

4.1. Recovery sequence in Cl(Ω;Rⁿ). Consider u∈Cl(Ω;Rⁿ) and fix once and for all a unitary vector field ν = νu which is normal Hⁿ⁻¹-a.e. to K = Ju∩Ω. Notice that, since J_u is the finite union of closed and pairwise disjoint (n−1)-dimensional simplexes, then the point where ν is not well-defined is a set of dimension at most n−2. The projection operator P : Ω→K is well defined almost everywhere around a small tubular neighborhood T ⊂Ω of K and thus we can consider, for points in T, thesigned distance

dist(x, K) = (x−x)·ν(x), x=P(x).

We consider a normal extension of ν on T. We now introduce the recovery sequence. Set ϑ:J_u→R, a function such that

ϑ∈W₀^1,∞(J_u;R,Hⁿ⁻¹), ϑ >0 onJ_u,

to be chosen later. We also require that ϑ(x) = 0 for allx∈K\J_u. For any ε >0 small enough, consider the set defined as

A_ϑε :={y+tν(y) |y∈J_u, t∈(−ϑ(y)ε, ϑ(y)ε)}.

Notice that up to choose ε small enough it is not restrictive to assume that A_ϑε has finitely many disconnected component well separated one from another, each of which is part of a tubular neighborhood of an (n−1)-dimensional hyperplane (see Figure 4.1).

Indeed, as explained briefly in Remark 4.1, up to carefully removing the singularity of the simplex where Ju lives and extending u smoothly on the cut (or by arguing just in the case where Ω is a cube and the jump set is an hyperplane as it is done in [27]), we obtain (asymptotically) the same result. Note that this machinery would only make the computations heavier without adding any relevant generality to our proof; thus we will avoid it. With the same carefulness (or by suitably modify the construction provided by Theorem4.8, see [27, Remark 3]), it is not restrictive to assume alsoK =Ju∩Ω⊂Ω.

Remark 4.1. LetH₁, H₂ be two hyperplanes such thatJ_u ⊂H₁∩H₂. Then consider the tubular neighborhood given by the Minkowski sumT1,2(ε) :=H1∩H2+Bε. Assume that we are able to define our recovery sequence u_ε, v_ε for any x ∈ Ω\T_1,2(ε). Then we can extend it to an H¹(Ω;Rⁿ)×V_ε pair (u_ε, v_ε) on Ω through the solution of the 2-capacity

problem inT_1,2(ε) (see [20, proof of Corollary 3.11]). In particular the contribution to the energy of the pairs (uε, vε) on the set T1,2(ε) is given by

F_ε(uε, vε;T1,2(ε)) = Z

T1,2(ε)

Ae(uε·e(uε) +F(x, e(uε, vε) +ψ(v_ε) ε

dx

≤C

"

Z

T1,2(ε)

|∇u_ε|²dx+ |T_1,2(ε)|

ε

# .

Since the 2-capacity of H1∩H2 inT1,2(ε) is 0 and since |T_1,2(ε)|/ε →0 as ε approaches 0 we can conclude that the contribution to the energy of such pairs, on the set T_1,2(ε), is asymptotically negligible. For this reason in the sequel we will assume without loss of generality that the jump set is always contained in the pairwise disjoint union of pieces of hyperplane (see Figure4.1).

Having in mind this additional assumption on the jump set, we define the following functions

vε(x) =











1 ifx /∈A_(ϑ+1)ε

1−αε ε

|dist(x, J_u)| −ϑ(x) + (1 +ϑ(x))αε ifx∈A_(ϑ+1)ε\A_ϑε

αε ifx∈A_ϑε

(4.2) and

u_ε(x) =



















u ifx /∈Aϑε

u(x+ϑ(x)εν)−u(x−ϑ(x)εν) 2ϑ(x)ε

dist(x, J u)

+u(x+ϑ(x)εν) +u(x−ϑ(x)εν) 2

ifx∈Aϑε

. (4.3)

Remark 4.2 (On the regularity of (u_ε, v_ε)). When xapproachesJ_u\J_u we haveu_ε(x) = u(x) and thus we can conclude uε ∈ W^1,∞(Ω;Rⁿ). On the other hand, we see that vε

might present a jump on the lines

{y+tν |y∈J_u\J_u, t∈(−ε, ε)}

whereϑ(y) = 0. To overcome this problem we can argue as follows. As a consequence of [20, Corollary 3.11, Assertion ii”)] we can claim that the better regularity of the jump set of u ensures thatHⁿ⁻²(Ju\Ju)<+∞ and thus, for every ε >0 we can cover such a set with a finite numberN_ε of balls B_k(ε) of radius εsuch that limε→0N_εεⁿ⁻¹= 0.

Moreover, we can find a function ζε such that ζε = 1 outside Σε := SNε

k=1Bk(2ε),

|∇ζ_ε| ≤ 1/ε, ζ_ε = αε on ∪_kB_k(ε). In particular we can make use of the neighbourhood SNε

k=1B_k(3ε) \Σ_ε to sew up ζ_ε1Σε with v_ε(1−1Σε) in an H¹ way. Furthermore, the slope of the function constructed in this way can be controlled by 1/ε and hence the gradient of the surgery, namely ˆv_ε, still has modulus less than 1/ε(up to the carefulness of Remark 4.3) as required by the constraint. In particular, by considering ˆvε in place of v_ε we can see that ˆv_ε ∈ V_ε. In order to alleviate the notations we will neglect this correction that, indeed, does not affect the energy asymptotically, due to the fact that

|SNε

k=1B_k(3ε)|/ε≤CNεεⁿ⁻¹ →0.

Remark 4.3 (On the constraint|∇v_ε| ≤1/ε). Notice that

|∇v_ε|= (1−αε) ε

p[1 +ε²|∇ϑ(x)|²]≤Cε/ε

where Cε & 1. In particular we can correct our vε by dividing by the factor Cε >1 so to ensure|∇v_ε| ≤1/εwithout essentially changing the structure of the recovery sequence.