Sharp upper bounds for a singular perturbation problem related to micromagnetics

Sharp upper bounds for a singular perturbation problem related to micromagnetics

ARKADYPOLIAKOVSKY

Abstract. We construct an upper bound for the following family of functionals {E_ε}_ε>0, which arises in the study of micromagnetics:

E_ε(u)=

ε|∇u|²+1 ε

R²|Hu|².

Hereis a bounded domain inR²,u∈ H¹(,S¹)(corresponding to the mag- netization) andH_u, the demagnetizing field created byu, is given by

div(u˜+H_u)=0 inR², curlH_u=0 inR²,

whereu˜is the extension ofuby 0 inR²\. Our upper bound coincides with the lower bound obtained by Rivi`ere and Serfaty.

Mathematics Subject Classification (2000): 49J45 (primary); 35B25, 35J20 (secondary).

1.

Introduction

In this paper we study the following energy-functional, related to micromagnetics:

E_ε(u):=

ε|∇u|²+1 ε

R²|H_u|². (1.1) Here is a bounded domain in

R

² with Lipschitz boundary, u is a unit-valued vector-field (corresponding to the magnetization) in H¹(,S¹) and H_u, the de- magnetizing field created byu, is given by

div(u˜+H_u)=0 in

R

²

curlH_u =0 in

R

², (1.2)

Received December 5, 2006; accepted in revised form November 22, 2007.

whereu˜is the extension ofuby 0 in

R

^2\. For the physical models related toE_ε, we refer to [18] and all the references therein.

We can rewrite (1.1) in the following form. Denoting by⁻¹u˜ the Newtonian potential ofu, we observe that the vector-field˜ H¯_u := −∇

div(⁻¹u˜)

belongs to L²(

R

²,

R

²). Moreover,

divH¯_u= −divu˜ in

R

² curlH¯_u =0 in

R

². SoH_u = ¯H_uand we obtain

E_ε(u)=

ε|∇u|²+ 1 ε

R²∇

div(⁻¹u˜)². (1.3) In [19] T. Rivi`ere and S. Serfaty proved the following theorem, giving compactness and a lower bound for the energies E_ε.

Theorem 1.1. Letbe a bounded simply connected domain in

R

^{2. Letεn → 0} and u_n ∈ H¹(,S¹)with a liftingϕn ∈ H¹(,

R

^{)i.e., un = eiϕn} a.e., and such that

E_εn(u_n)≤C (1.4)

ϕnL^∞()≤N. (1.5)

Then, up to extraction of a subsequence, there exists u andϕin∩q<∞L^q()such that

ϕn →ϕin ∩q<∞L^q() u_n →u in ∩q<∞L^q() . Moreover, if we consider

T^tϕ(x):=inf ϕ(x),t T^tu(x):=e^{i Ttϕ(x)},

thendiv_xT^tu is a bounded Radon measure on×

R

^{, with t →divxTt}u continuous from

R

^to

D

(). In addition

2

R

div_xT^tud x dt≤ lim

n→∞

2|∇ϕn·H_un| ≤ lim

n→∞E_εn(u_n) <∞. The main contribution of this paper is to establish the upper bound for E_ε in the case whereuand its liftingϕbelong toB V. First of all we observe that ifεn →0,

E_εn(u_n) ≤ C, and u_n → u in L^q, where |u_n| = 1 and u ∈ B V then clearly

nlim→∞

R²|H_un|²=0, which implies

nlim→∞

R²u˜_n· ∇δ= − lim

n→∞

R²H_un· ∇δ=0 ∀δ∈C_c^∞(

R

^2,

R

^{) .} Therefore, divu˜ =0 as a distribution,i.e.,







|u| =1 a.e. in , divu=0 in , u·n=0 on∂ .

(1.6)

The main result of this paper is the following theorem.

Theorem 1.2. Let be a bounded domain in

R

² with Lipschitz boundary. Con- sider u ∈ B V(,S¹), satisfyingdivu =0inand u·n=0on∂and assume there existϕ ∈ B V(,

R

⁾, such that u= e^iϕ a e. in. Then there exists a family of functions{v_ε} ⊂C²(

R

^N,

R

^)satisfying

ε→lim0⁺v_ε(x)=ϕ(x) in L¹(,

R

⁾ and

ε→lim0E_ε(e^ivε)=2

R

div_xT^tud x dt. Moreover, ifϕ∈ B V(,

R

^)∩L∞, then we have

ε→lim0⁺v_ε(x)=ϕ(x) in L^p(,

R

^{) ∀p∈ [1,∞) .}

In order to construct{v_ε}we take the convolution ofϕ with a varying smoothing kernel, i.e., we setv_ε(x) := ε⁻²

R²η_y−x

ε ,x

ϕ(y)d y where η ∈ C_c²(

R

^2×

R

²⁾ satisfies

R²η(z,x)d x = 1 for every x ∈ , and we optimize the choice of the kernel η. A similar approach was used in [16] and [17], but a new ingredient is required here, since the non-local term

R²|H_u|²gives certain difficulties.

1.1. The basic idea

We shall follow essentially the strategy of [16] and [17]. The main new ingredient here is the calculation of

ε→lim0⁺

1 ε

R²

∇

div(⁻¹{χe^ivε})². (1.7) We first calculate

L(l):= lim

ε→0⁺

1 ε

R²∇

div(⁻¹{ϕ_ε})²,

where ϕ_ε(x):=ε⁻²

R²l_y−x

ε ,x

ϕ(y)d y, with l ∈ C_c²(

R

^{2 ×,}

R

^2)satisfying

R²l(z,x)d x =0 for every x ∈. Since∇

div(⁻¹{φ_ε})

has the same asymp- totic behavior asε⁻²

R²s_y−x

ε ,x

ϕ(y)d y, wheres(z,x):=∇z

div_z(⁻_z¹l(z,x)) , we can calculate the limit L(l)in a similar way to what was done in [16] and [17]

(see Lemmas 3.1 and 3.2 for the details). Using the results of [17] (see Proposition 2.2 below) it is easy to calculate

D(l):= lim

ε→0

1 ε

e^ivε −ϕ_ε−u²d x.

Now, given a fixed η and a small δ > 0, we choosel = l_δ in such a way that D(l_δ) < δ. Then, using the estimate

R²∇

div(⁻¹{χf})²≤C

|f|², we deduce that

ε→lim0

1 ε

R²∇

div(⁻¹{χe^ivε})²−1 ε

R²∇

div(⁻¹{χϕ_ε})²

≤ lim

ε→0C 1

ε

R²

∇

div(⁻¹{χ(e^ivε−ϕε−u)})²1/2

≤Cδ^1/2. Finally, tettingδ tend to 0, we conclude that the limit in (1.7) should be equal to lim_δ→0L(l_δ). We follow basically this strategy in the proof of Proposition 4.1.

ACKNOWLEDGEMENTS. I am grateful to Prof. Camillo De Lellis for proposing this problem to me and for some useful suggestions. Part of this research was done dur- ing a visit at the Laboratoire J. L. Lions of the University Paris VI in the framework of the RTN-Programme Fronts-Singularities. I am indebted to Prof. Haim Brezis for the invitation and for many stimulating discussions.

2.

Preliminaries

Throughout this section we assume thatis a bounded domain in

R

²with Lipschitz boundary. We begin by introducing some notation. For every ν ∈ S¹ (the unit sphere in

R

^{2) andR}>0 we denote

B⁺_R(x,ν)= {y ∈

R

^{2 : |y−x|< R, (y−x)·ν >0}, (2.1)} B⁻_R(x,ν)= {y ∈

R

^{2: |y−x|< R, (y−x)·ν <0}, (2.2)} H₊(x,ν)= {y ∈

R

^{2:(y−x)·ν >0}, (2.3)} H₋(x,ν)= {y ∈

R

^{2:(y−x)·ν <0} (2.4)}

and

H_ν⁰= {y ∈

R

^{2 : y·ν=0}. (2.5)}

Definition 2.1. Consider a function f ∈B V(,

R

^m)and a pointx ∈.

i) We say thatx is a point ofapproximate continuityof f if there exists z ∈

R

^m such that

ρ→lim0⁺

B_ρ(x)|f(y)−z|d y

L

^2Bρ(x) =0.

In this casezis called anapproximate limitof f atxand we denotezby f˜(x). The set of points of approximate continuity of f is denoted byG_f.

ii) We say thatx is anapproximate jump pointof f if there exista,b ∈

R

^{m and} ν ∈S^N−1such thata=band

ρ→lim0⁺

B_ρ⁺(x,ν)|f(y)−a|d y

L

^2Bρ(x) =0, lim

ρ→0⁺

B_ρ⁻(x,ν)|f(y)−b|d y

L

^2Bρ(x) =0. (2.6) The triple (a,b,ν), uniquely determined by (2.6) up to a permutation of (a,b) and a change of sign of ν, is denoted by(f⁺(x), f⁻(x),νf(x)). We shall call νf(x)the approximate jump vectorand we shall sometimes write simplyν(x)if the reference to the function f is clear. The set of approximate jump points is denoted by J_f. A choice of ν(x)for everyx ∈ J_f (which is unique up to sign) determines an orientation of J_f. At a point of approximate continuity x, we shall use the convention f⁺(x)= f⁻(x)= ˜f(x).

We refer to [2] for the results on BV-functions that we shall use in the sequel.

Consider a function = (ϕ1, ϕ2, . . . , ϕd) ∈ B V(,

R

^d). By [2, Proposi- tion 3.21] we may extend to a function¯ ∈ B V(

R

^2,

R

^d), such that¯ = a.e. in , supp¯ is compact and D(∂)¯ = 0. From the proof of Proposi- tion 3.21 in [2] it follows that if∈ B V(,

R

^d)∩L∞then its extension¯ is also in B V(

R

^2,

R

^d)∩L∞. Consider also a matrix valued function∈C_c²(

R

^2×

R

^2,

R

^l×d) For everyε >0 define a function_ε(x):

R

^2→

R

^{l by}

_ε(x):= 1 ε²

R²y−x ε ,x

· ¯(y)d y

=

R²(z,x)· ¯(x+εz)d z, ∀x ∈

R

^2.

(2.7)

Thanks to [17, Proposition 3.2], we have the following result. It generalizes Propo- sition 3.2 from [16] and provides the key tool for the calculation of the upper bound, both in [17] and in the current paper. In the proof of Lemma 3.2 we shall also follow the general strategy of its proof in [17].

Proposition 2.2. Let W ∈C¹(

R

^l×

R

^q,

R

^)satisfying

∇aW(a,b)=0 whenever W(a,b)=0. (2.8) Consider∈B V(,

R

^{d)∩L∞and u ∈B V(,}

R

^{q)∩L∞satisfying}

W R²(z,x)d z

·(x), u(x)

=0 for a.e. x∈ , where∈C_c²(

R

^2×

R

^2,

R

^l×d), as above. Let_ε be as in(2.7). Then,

ε→lim0

1 εW

_ε(x),u(x) d x

=

J

₀

−∞W

(t,x),u⁺(x) dt +

_+∞

0

W

(t,x),u⁻(x) dt

d

H

¹(x) , (2.9)

where

(t,x)= ^t

−∞P(s,x)ds

·⁻(x)+ ^+∞

t

P(s,x)ds

·⁺(x), (2.10) with

P(t,x)=

H_ν(x)⁰

(tν(x)+y,x)d

H

¹(y) , (2.11) ν(x)is the jump vector ofand it is assumed that the orientation of J_u coincides with the orientation of J

H

^{1a.e. on J}u∩J.

Definition 2.3. Given f ∈ L^∞(

R

^2,

R

^k)with compact support, we define its New- tonian potential

⁻¹f (x):=

R²

1

2π ln|x−y| f(y)d y. Than it is well known that

R²∇^{2 −1}f

(x)²d x =

R²|f(x)|²d x, (2.12) where givenv=(v1, . . . , vk):

R

^{2 →}

R

^k we denote by∇²v∈

R

^{k×2×2the tensor} withli j-th component∂_{i j}²vl.

Definition 2.4. Let

V

be the class of all functionsη∈C_c²(

R

^2×

R

^2,

R

^{)such that}

R²η(z,x)d z=1 ∀x ∈ . (2.13) Let

U

be the class of all functionsl(z,x)∈C_c²(

R

²×,

R

²)such that

R²l(z,x)d z=0 ∀x∈

R

². (2.14)

In [17, Lemma 5.1], we proved the following statement. This statement generalize Claim 3 of Lemma 3.4 from [16] and was an essential tool in the optimizing the upper bound in [17].

Lemma 2.5. Letµ be positive finite Borel measure on andν0(x) : →

R

² a Borel measurable function with |ν0| = 1. Let

W

¹ denote the set of functions

p(t,x):

R

^×→

R

satisfying the following conditions:

i) p is Borel measurable and bounded,

ii) there exists M >0such that p(t,x)=0for|t|> M and any x ∈, iii)

R p(t,x)dt =1,∀x ∈.

Then for every p(t,x) ∈

W

1, there exists a sequence of functions{ηn} ⊂

V

^(see Definition2.4), such that the sequence of functions{p_n(t,x)}defined on

R

^×by

p_n(t,x)=

H_ν⁰ 0(x)

ηn(tν0(x)+y,x)d

H

¹(y),

has the following properties:

i) there exists C₀such thatp_nL^∞ ≤C₀for every n,

ii) there exist M >0such that p_n(t,x)=0for|t|>M and every x ∈, for all n.

iii) lim_n→∞

R|p_n(t,x)− p(t,x)|dt dµ(x)=0.

With the same method it is not difficult to prove

Lemma 2.6. Letµ be positive finite Borel measure on andν0(x) : →

R

² a Borel measurable function with |ν0| = 1. Let

W

0 denote the set of functions q(t,x):

R

^×→

R

²satisfying the following conditions:

i) q is Borel measurable and bounded,

ii) there exists M >0such that q(t,x)=0for|t|> M and every x∈. iii)

Rq(t,x)dt =0, ∀x ∈.

Then for every q(t,x) ∈

W

⁰, there exists a sequence of functions{l_n} ⊂

U

^(see Definition2.4), such that the sequence of functions{q_n(t,x)}defined on

R

^×by

q_n(t,x)=

H_ν⁰ 0(x)

l_n(tν0(x)+y,x)d

H

¹(y), has the following properties:

i) there exists C0such thatqnL^∞ ≤C0for every n,

ii) there exist M >0such that q_n(t,x)=0for|t|> M and every x ∈, for all n,

iii) lim_n→∞

R|q_n(t,x)−q(t,x)|dt dµ(x)=0.

3.

First estimates

Throughout this section we assume thatis a bounded domain in

R

²with Lipschitz boundary.

Letl ∈

U

(see Definition 2.4). Considerr(z,x) := ⁻_z¹l(z,x). Then r ∈ C²(

R

^{2 ×}

R

^2,

R

^{2) with suppr ⊂}

R

^{2 × K, where K}

. Moreover, since

R²l(z,x)d z=0, for everyk=0,1,2. . .we have the estimates

|∇^k_xr(z,x)| ≤ C_k

|z| +1, ∇_x^k

∇zr(z,x)≤ C_k

|z|²+1, ∇^k_x

∇²_zr(z,x)≤ C_k

|z|³+1,

(3.1)

whereC_k >0 does not depend onzandx.

Lemma 3.1. Letϕ ∈ B V(,

R

^{)∩L∞and l ∈}

U

(see Definition2.4). For every ε >0consider the functionϕ_ε ∈C¹(

R

^2,

R

^2)by

ϕ_ε(x):= 1 ε²

R²l y−x

ε ,x

ϕ(¯ y)d y =

R²l(z,x)ϕ(¯ x+εz)d z, (3.2) where ϕ¯ is some bounded B V extension ofϕ to

R

² with compact support. Next consider r(z,x):=⁻_z¹l(z,x)and set

ξ_ε(x):= 1 ε²

R²∇1

div₁ry−x ε ,x

ϕ(¯ y)d y

=

R²∇z

div_zr(z,x)

¯

ϕ(x+εz)d z,

(3.3)

where∇1(div₁r)is the gradient of divergence of r(z,x)in its first variable, namely z. Then,

R²

1 ε∇

div(⁻¹ϕ_ε)

(x)²d x =o_ε(1)+

1

εϕ_ε(x)·ξ_ε(x)d x. (3.4) Proof. Sincel(z,x)=0 ifx∈/ K, whereK is some compact subset of, we have, in particular,ϕ_ε(x)=0 for everyx∈

R

^2\. Then, integrating by part two times, we conclude

R²

1 ε∇

div(⁻¹ϕε) (x)²d x

= −

R²

1 ε

div(⁻¹ϕ_ε) (x)·

div(⁻¹ϕ_ε) (x)d x

= −

R²

1

εdivϕ_ε(x)·

div(⁻¹ϕ_ε) (x)d x

=

1

εϕ_ε(x)· ∇

div(⁻¹ϕ_ε) (x)d x.

(3.5)

Next consider the functionζ_ε ∈C¹(

R

^2,

R

^{2)given by} ζ_ε(x):= 1

ε²

R²r y−x

ε ,x

ϕ(¯ y)d y=

R²r(z,x)ϕ(¯ x+εz)d z. (3.6) We will prove now that

ε²∇²ζ_ε(x)−

R²∇_z²r(z,x)ϕ(¯ x+εz)d z

≤Cε^2/3 ∀x ∈ . (3.7) We shall denote by∇1l and∇2lthe gradient ofl(z,x)with respect to the variables zandx respectively. We have,

ε²∇²ζ_ε(x)−

R²∇z²r(z,x)ϕ(¯ x+εz)d z

=

R²∇x²r y−x

ε ,x

ϕ(¯ y)d y− 1 ε²

R²∇₁²r y−x

ε ,x

ϕ(¯ y)d y

= −1 ε

R²

∇1∇2r y−x

ε ,x

+ ∇2∇1r y−x

ε ,x

ϕ(¯ y)d y

+

R²∇₂²r y−x

ε ,x

ϕ(¯ y)d y.

(3.8)

Therefore, by the H¨older inequality and the estimates in (3.1), we obtain ε²∇²ζ_ε(x)−

R²∇_z²r(z,x)ϕ(¯ x+εz)d z

≤ε^2/3 1

ε²

R²

∇1∇2r y−x

ε ,x

+∇2∇1r y−x

ε ,x ^6/d y⁵

_5/6

R²| ¯ϕ(y)|⁶d y _1/6

+ε^2/3 1

ε²

R²

∇₂²r y−x

ε ,x³d y 1/3

R²| ¯ϕ(y)|^3/2d y 2/3

=ε^2/3

R²

∇1∇2r z,x

+ ∇2∇1r

z,x^6/5d z 5/6

R²| ¯ϕ(y)|⁶d y 1/6

+ε^2/3

R²

∇₂²r

z,x³d z 1/3

R²| ¯ϕ(y)|^3/2d y 2/3

≤Cε^2/3 which gives (3.7). In particular,

ε²ζ_ε(x)−ϕ_ε(x) =

ε²ζ_ε(x)−

R²zr(z,x)ϕ(¯ x+εz)d z

≤C₀ε^2/3.

(3.9)

Next by (3.5),

R²

1 ε∇

div(⁻¹ϕ_ε)

(x)²d x=

1

εϕ_ε(x)· ∇

div(⁻¹ϕ_ε) (x)d x

=

1

ε ϕ_ε(x)· ∇ div

ε²ζ_ε(x) d x

−

1

ε ϕ_ε(x)· ∇ div

⁻¹

ε²ζ_ε−ϕ_ε (x)d x.

(3.10) The last integral can be estimated by

1

εϕ_ε(x)· ∇ div

⁻¹

ε²ζ_ε−ϕ_ε (x)d x

≤

1

ε|ϕ_ε(x)|² 1/2

1 ε

R²

∇ div

⁻¹

ε²ζ_ε−ϕ_ε (x)²d x

1/2

≤

1

ε|ϕ_ε(x)|² 1/2

2 ε

R²

∇^{2 −1}

ε²ζ_ε−ϕ_ε (x)²d x

1/2

=

1

ε|ϕ_ε(x)|² _1/2

2

εε²ζ_ε(x)−ϕ_ε(x)²d x _1/2

. Then, since

1

ε|ϕ_ε(x)|²≤C

1 ε|ϕ_ε(x)|

≤ ¯C

1 ε

B_R(0)l(z,x)

¯

ϕ(x+εz)−ϕ(x) d z

d x

≤ ¯C

B_R(0)

1

ε|l(z,x)|

ϕ(¯ x+εz)−ϕ(x)d x d z

≤ ¯CDϕ(¯

R

²⁾

BR(0)|l(z,x)| · |z|d z= O(1) ,

(3.11)

using (3.9), from (3.10) we infer

R²

1 ε∇

div(⁻¹ϕ_ε)

(x)²d x=o_ε(1)+

1

εϕ_ε(x)·∇

div

ε²ζ_ε(x)

d x. (3.12) Next we remind thatξ_ε is defined by (3.3). By (3.7), we have,

∇ div

ε²ζ_ε(x)

−ξ_ε(x)≤ ¯Cε^2/3 ∀x ∈ . (3.13)

Then as before,

1 εϕ_ε(x)·

∇ div

ε²ζ_ε(x)

−ξ_ε(x)

d x

≤

1

ε|ϕ_ε(x)|² _1/2

1 ε∇

div

ε²ζ_ε(x)

−ξ_ε(x)²d x _1/2

≤Cε^1/6.

Therefore, from (3.12) we infer (3.4).

Lemma 3.2. Letϕ ∈ B V(,

R

^{)∩L∞and l ∈}

U

(see Definition2.4). For every ε >0and every x∈

R

²consider the functionϕ_ε ∈C¹(

R

^2,

R

^{2)as in}(3.2). Then,

ε→lim0

R²

1 ε∇

div(⁻¹ϕ_ε) (x)²d x

=

J_ϕ

_+∞

−∞ |ϕ⁺(x)−ϕ⁻(x)|²· ν(x)·

_+∞

t

q(s,x)ds ²dt

d

H

^{1(x) ,} (3.14)

where

q(t,x)=

H_ν(x)⁰

l(tν(x)+y,x)d

H

¹(y) , (3.15) andν(x)is the jump vector ofϕ.

Proof. By Lemma 3.1 we have

R²

1 ε∇

div(⁻¹ϕ_ε)

(x)²d x =o_ε(1)+

1

εϕ_ε(x)·ξ_ε(x)d x. (3.16) From this point the strategy of the proof is similar to that in [16] and [17] (see also Proposition 2.2). The only difference is that here ξ_ε is defined by a convolution of ϕ¯ with a kernel whose support is not compact. However, it turns out that this difference is not crucial and we can use almost the same approach.

Step 1. We prove a useful expression,

1

εϕ_ε(x)·ξ_ε(x)d x

= ₁

0 R²

1 t²ε²

∩B_Rtε(y)

ξ_ε(x)·l y−x

tε ,x y−x

tε d x

·d[Dϕ](¯ y)

dt, (3.17)

where ξ_ε is as in (3.3) and R > 0 is such thatl(z,x) = 0 whenever|z| ≥ R.

As before, we shall denote by ∇1l and∇2l the gradient ofl with respect to the first and second variables respectively. Denote

ϕtε,1(x), ϕtε,2(x)

:= ϕtε(x)and l₁(z,x),l₂(z,x)

:=l(z,x). Then for everyt ∈(0,1], every j ∈ {1,2}and every x∈

R

^{2we have}

d

ϕtε,j(x)

dt = d

dt 1

t²ε²

R²l_j y−x

tε ,x

ϕ(¯ y)d y

=− 1 t³ε²

R²

∇1l_j y−x

tε ,x · y−x

tε +2l_j y−x

tε ,x

ϕ(¯ y)d y

= − 1 t²ε

R²div_y

l_j y−x

tε ,x y−x

tε

¯ ϕ(y)d y

= 1 t²ε

R²l_j y−x

tε ,x y−x

tε ·d[Dϕ](¯ y) .

(3.18)

Therefore, for anyρ∈(0,1)we have,

1 ε

ϕ_ε(x)−ϕ_ρε(x)

·ξ_ε(x)d x

=

1

εξ_ε(x)· ¹

ρ

d ϕtε(x)

dt

d x

=

1 ρ ξ_ε(x)·

1 t²ε²

R²l y−x

tε ,x

y−x

tε ·d[Dϕ](¯ y) dt

d x

= ₁

ρ ξ_ε(x)· 1

t²ε²

R²l y−x

tε ,x

y−x

tε ·d[Dϕ](¯ y) d x

dt

= ₁

ρ R²

1 t²ε²

∩B_Rtε(y)

ξ_ε(x)·l y−x

tε ,x y−x

tε d x

·d[Dϕ](¯ y)

dt. (3.19)

From our assumptions onϕ, by (3.1), it follows that there exists a constantC >0, independent ofρ, such thatξ_ρ(x)≤C for everyρ >0 and everyx ∈. There- fore, lettingρtend to zero in (3.19), using the fact that lim_ρ→0ϕ_ρ(x)L¹() = 0 (see (3.11)), we get (3.17).

Step 2. We prove the identity

R²

1 ε∇

div(⁻¹ϕ_ε) (x)²d x

=o_ε(1)+ 1

0 J_ϕ B_R(0)

l(z,x)·ξ_ε(x−εt z) z d z

·d Dϕ

(x)

dt

+ 1

0 G_ϕ B_R(0)

l(z,x)·ξ_ε(x−εt z) z d z

·d Dϕ

(x)

dt,

(3.20)

whereG_ϕis the set of approximate continuity ofϕ. By (3.16) and (3.17) we deduce

R²

1 ε∇

div(⁻¹ϕ_ε) (x)²d x

=o_ε(1)+ ₁

0 R²

1 t²ε²

∩B_Rtε(y)

ξ_ε(x)·l y−x

tε ,x y−x

tε d x

·d[Dϕ](¯ y)

dt

=o_ε(1)+ ₁

0 R²

1 t²ε²

K∩B_Rtε(y)

ξ_ε(x)·l y−x

tε ,x

y−x tε d x

·d[Dϕ](¯ y)

dt, (3.21) whereK

is a compact set (see Definition 2.4). But, for everyε <_R¹dist(K, ∂) we have

R²

1 t²ε²

K∩B_Rtε(y)

ξ_ε(x)·l y−x

tε ,x

y−x tε d x

·d[Dϕ](¯ y)

=

1 t²ε²

K∩B_Rtε(y)

ξ_ε(x)·l y−x

tε ,x

y−x tε d x

·d[Dϕ](¯ y)

=

1 t²ε²

R²

ξ_ε(x)·l y−x

tε ,x

y−x tε d x

·d[Dϕ](¯ y) .

Therefore, by (3.21), we obtain

R²

1 ε∇

div(⁻¹ϕ_ε) (x)²d x

=o_ε(1)+ 1 0

1 t²ε²

R²

ξ_ε(x)·l y−x

tε ,x

y−x tε d x

·d[Dϕ](¯ y)dt

=o_ε(1)+ 1

0 B_R(0)

l(z,y−εt z)·ξ_ε(y−εt z) z d z

·d Dϕ

(y)

dt

=o_ε(1)+ 1

0 B_R(0)

l(z,x)·ξ_ε(x−εt z) z d z

·d Dϕ

(x)

dt,

(3.22)

where in the last equality we used the estimate|l(z,x−εt z)−l(z,x)| ≤Cεt|z|. Therefore we obtain (3.20).

Step 3. We will prove that the second integral in the r.h.s of (3.20) vanishes as ε→0. For everyxinG_ϕwe have,

ρ→lim0⁺

1 ρ²

B_ρ(x)| ¯ϕ(y)− ˜¯ϕ(x)|d y =0. Takingρ= Lε, for everyL >0, gives

ε→lim0⁺

B_L(0)| ¯ϕ(x+εz)− ˜¯ϕ(x)|d z=0, forxinG_ϕ. (3.23) Using (3.1), since

R²∇z

div_zr(z,x−εt y)

d z =0, for everyxinG_ϕ,y∈ B_R(0), t ∈ [0,1]andL>0 we have,

|ξ_ε(x−εt y)| =

R²∇z

div_zr(z,x−εt y)

¯

ϕ(x+εz−εt y)d z

=

R²∇z

div_zr(z,x−εt y)

¯

ϕ(x+εz−εt y)− ˜¯ϕ(x) d z

≤

BL(0)

∇z

div_zr(z,x−εt y)·ϕ(¯ x+εz−εt y)− ˜¯ϕ(x)d z +

R²\B_L(0)

∇z

div_zr(z,x−εt y)·ϕ(¯ x+εz−εt y)− ˜¯ϕ(x)d z

≤A_L

BL(0)| ¯ϕ(x+εz−εt y)− ˜¯ϕ(x)|d z+B

R²\BL(0)

1

|z|³+1d z

≤A_L

B_(L+R)(0)| ¯ϕ(x+εz)− ˜¯ϕ(x)|d z+B

R²\BL(0)

1

|z|³+1d z,

(3.24)

where B > 0 is a constant and A_L > 0 depends only on L. Givenδ > 0 we can takeL>0 such that

B

R²\BL(0)

1

|z|³+1d z < δ ,

Then, using (3.24) and (3.23), we infer lim_ε→0+|ξ_ε(x−εt y)|< δand sinceδwas arbitrary,

ε→lim0⁺ξ_ε(x−εt y)=0 ∀x ∈G_ϕ, y∈ B_R(0), t ∈ [0,1]. (3.25) Using (3.1), we also have|ξ_ε(x −εt y)| ≤C, and therefore, plugging (3.25) into (3.20), we obtain

R²

1 ε∇

div(⁻¹ϕ_ε) (x)²d x

=o_ε(1)+ 1

0 J_ϕ B_R(0)

l(z,x)·ξ_ε(x−εt z) z d z

·d Dϕ

(x)

dt. (3.26)

Step 4. Considerl¯(z,x):= ∇z

div_zr(z,x)

. For everyε,t ∈ (0,1),x ∈ J_ϕ and z∈B_R(0), we have

ξ_ε(x−εt z)=

R²

l¯(y,x−εt z)ϕ¯

x+ε(y−t z) d y

=

R²

l¯(y+t z,x−εt z)ϕ¯ x+εy

d y

=

H₊(0,ν(x))

l¯(y+t z,x−εt z)ϕ¯ x+εy

d y

+

H₋(0,ν(x))

l¯(y+t z,x−εt z)ϕ¯ x+εy

d y

=

H₊(0,ν(x))

l¯(y+t z,x−εt z)ϕ⁺(x)d y

+

H₋(0,ν(x))

l¯(y+t z,x−εt z)ϕ⁻(x)d y

+

H₊(0,ν(x))

l¯(y+t z,x−εt z)

¯

ϕ(x+εy)−ϕ⁺(x) d y

+

H₋(0,ν(x))

l¯(y+t z,x−εt z)

¯

ϕ(x+εy)−ϕ⁻(x) d y.

(3.27)