The relaxed energy for <span class="mathjax-formula">${S}^{2}$</span>-valued maps and measurable weights

(1)

www.elsevier.com/locate/anihpc

The relaxed energy for S ² -valued maps and measurable weights L’énergie relaxée des applications à valeurs dans S ²

et poids mesurables

Vincent Millot

Laboratoire J.L. Lions, université Pierre et Marie Curie, B.C. 187, 4, place Jussieu, 75252 Paris cedex 05, France Received 26 October 2004; accepted 3 February 2005

Available online 10 May 2005

Abstract

We compute explicitly a relaxed type energy for mapsu:Ω⊂R³→S². The explicit formula involves the length of a minimal connection relative to some specific distance connecting the topological singularities ofuand associated to a measurable weight function. This result generalizes a previous result of F. Bethuel, H. Brezis and J.M. Coron.

Résumé

Nous calculons explicitement une énergie de type relaxée pour des applicationsu:Ω⊂R³→S². La formule explicite fait intervenir la longueur d’une connexion minimale relative à une certaine distance, connectant les singularités topologiques de uet associée à une fonction de poids mesurable. Ce résultat généralise un résultat antérieur de F. Bethuel, H. Brezis et J.M.

Coron.

MSC: 49D20; 49F99

1. Introduction and main results

LetΩ be a smooth bounded and connected open set ofR³and letw:Ω→Rbe a measurable function such that

0< λwΛ a.e. inΩ (1.1)

E-mail address: millot@ann.jussieu.fr (V. Millot).

doi:10.1016/j.anihpc.2005.02.003

(2)

for some constantλandΛ. We setH_g¹(Ω, S²)= {u∈H¹(Ω, S²), u=gon∂Ω}, whereg:∂Ω→S²is a given smooth boundary data such that deg(g)=0. Our main goal in this paper is to obtain an explicit formula for the relaxed functional

E_w(u)=Inf

lim inf

n→+∞

Ω

∇u_n(x)²w(x)dx, u_n∈H_g¹(Ω, S²)∩C¹(Ω), u _n uweakly inH¹

,

defined foru∈H_g¹(Ω, S²). By a result of F. Bethuel (see [1]),H_g¹(Ω, S²)∩C¹(Ω) is sequentially dense for the weak topology inH_g¹(Ω, S²)and then the functionalE_wis well defined.

In [4], F. Bethuel, H. Brezis and J.M. Coron have proved that forw≡1, E1(u)=

Ω

∇u(x)²dx+8π L(u),

where L(u) denotes the length of a minimal connection relative to the Euclidean geodesic distance d_Ω in Ω connecting the singularities ofu(see also M. Giaquinta, G. Modica, J. Souˇcek [12]). Ifu∈H_g¹(Ω, S²)is smooth onΩexcept at a finite number of points inΩ, the length of a minimal connection relative tod_Ω connecting the singularities ofuis given by

L(u)= Min

σ∈SK

K i=1

d_Ω(P_i, N_{σ (i)}),

where(P₁, . . . , P_K)and(N₁, . . . , N_K)are respectively the singularities of positive and negative degree counted according to their multiplicity (since deg(g)=0, the number of positive singularities is equal to the number of negative ones) and SK denotes the set of all permutations ofK indices. For the definition ofL(u) when u is arbitrary in H_g¹(Ω, S²), we refer to (1.6), (1.7) below. The notion of length of a minimal connection between singularities has its origin in [8]. We also refer to the results of J. Bourgain, H. Brezis, P. Mironescu [5] and H. Brezis, P. Mironescu, A.C. Ponce [9] for similar problems involvingS¹-valued maps.

Foru∈H¹(Ω, S²), the vector fieldD(u)first introduced in [8] and defined by D(u)=

u· ∂u

∂x₂∧ ∂u

∂x₃, u· ∂u

∂x₃∧ ∂u

∂x₁, u· ∂u

∂x₁∧ ∂u

∂x₂

(1.2) plays a crucial role. Indeed, if u is smooth except at a finite number of points(Pi, Ni)^K_i₌₁ inΩ, then (see [8], Appendix B)

divD(u)=4π K i=1

(δ_P_i−δ_N_i) inD(Ω) (1.3)

and if in additionu_|∂Ω=g, we have (since deg(g)=0, see [8], Section IV) L(u)=Sup

_K

i=1

ζ (P_i)−ζ (N_i)

, (1.4)

where the supremum is taken over all functionsζ:Ω→Rwhich are 1-Lipschitz with respect to distancedΩ i.e.,

|ζ (x)−ζ (y)|dΩ(x, y). Note that for any real Lipschitz functionζ, K

i=1

ζ (P_i)−ζ (N_i)= 1 4π

Ω

divD(u)ζ= − 1 4π

Ω

D(u)· ∇ζ + 1 4π

∂Ω

D(u)·ν

ζ, (1.5)

(3)

whereν denotes the outward normal to ∂Ω. We recall thatD(u)·ν is equal to the 2×2 Jacobian determinant ofurestricted to∂Ω and then it only depends ong. In view of (1.4) and (1.5),L(u)has been defined in [4] for u∈H_g¹(Ω, S²)by

L(u)= 1 4πSup

T (u), ζ

, ζ:Ω→R1-Lipschitz with respect tod_Ω

, (1.6)

whereT (u)∈D(Ω)denotes the distribution defined by its action on real Lipschitz functions through the formula:

T (u), ζ

=

Ω

D(u)· ∇ζ −

∂Ω

D(u)·ν

ζ. (1.7)

In a previous paper [13], we have studied the following variational problem: given two distinct pointsP andN inΩ,

E_w(P , N )=Inf

Ω

∇v(x)²w(x)dx, v∈E(P , N )

,

where

E(P , N )=

v∈H¹(Ω, S²)∩C¹ Ω\ {P , N}

, v=const on∂Ω, T (v)=4π(δP −δN)inD(Ω) . In the casew≡1, H. Brezis, J.M. Coron and E. Lieb have shown that (see [8])

E₁(P , N )=8π dΩ(P , N ).

For an arbitrary functionw, we have proved (see [13]) thatE_w(·,·)defines a distance function satisfying

8π λd_Ω(·,·)E_w(·,·)8π Λd_Ω(·,·). (1.8)

From (1.8), we infer thatE_w extends toΩ× Ω into a distance onΩ. In what follows, we set for x, y∈ Ω, d_w(x, y)= 1

8πE_w(x, y).

Whenwis continuous, we also have shown that the distanced_wcan be characterized in the following way: for any x, y∈ Ω,

d_w(x, y)=Min 1

0

w γ (t )γ (t )˙ dt,

where the minimum is taken over all Lipschitz curve γ:[0,1] → Ω verifyingγ (0)=x andγ (1)=y. For an arbitrary measurable functionw, the previous formula is meaningless sincewis not well defined on curves but a similar characterization ofd_w actually holds. We refer to [13] for more details. We also recall the general result in [13]:

Theorem 1.1. Let(Pi)^K_i₌₁and(Ni)^K_i₌₁be two lists of points inΩand consider E (P_i, N_i)^K_i₌₁

=

v∈H¹(Ω, S²)∩C¹ Ω\

(P_i, N_i)^K_i₌₁ ,

v=const on∂ΩandT (v)=4π K i=1

δP_i−δN_i inD(Ω)

.

(4)

Then we have Inf

Ω

∇v(x)²w(x)dx, v∈E (P_i, N_i)^K_i₌₁

=8π L_w,

whereL_w is the length of a minimal connection relative to distanced_w connecting the points(P_i)and(N_i)i.e., Lw= Min

σ∈SK

K i=1

dw(Pi, Nσ (i)).

By analogy with the casew≡1, we define foru∈H_g¹(Ω, S²), L_w(u)= 1

4π Sup T (u), ζ

, ζ:Ω→R1-Lipschitz with respect tod_w

(note that any real functionζ which is 1-Lipschitz with respect tod_w, is a Lipschitz function with respect tod_Ω sinced_w is strongly equivalent tod_Ω and thenT (u), ζ is well defined). Whenu is smooth except at a finite number of points(P_i, N_i)^K_i₌₁inΩ, it follows as in [8] thatL_w(u)is equal to the length of a minimal connection relative to distanced_wconnecting the points(P_i)and(N_i). Our main result is the following.

Theorem 1.2. For anyu∈H_g¹(Ω, S²), we have Ew(u)=

Ω

∇u(x)²w(x)dx+8π Lw(u).

The proof of Theorem 1.2 is presented in Section 3 and is based on a method similar to the one used in [4]

and on a Dipole Removing Technique exposed in the next section. This technique is mostly inspired from [1] but involves some tools developed in [13] in order to treat the problem for a non smooth functionw.

In Section 4, we prove a stability property of E_w. More precisely, we give some conditions on a sequence (w_n)_n_∈Nunder which one can conclude that the sequence of functionals(E_w_n)_n_∈Nconverges pointwise toE_w on H_g¹(Ω, S²). The results are obtained using previous ones in [13]. In Section 5, we present similar results for a relaxed type functional in which we do not prescribed any boundary data.

Throughout the paper, a sequence of smooth mollifiers means any sequence(ρ_n)_n_∈Nsatisfying ρ_n∈C^∞(R³,R), Suppρ_n⊂B_1/n,

R³

ρ_n=1, ρ_n0 onR³.

2. The dipole removing technique

In this section, we first give a technical result which will be used for the dipole removing technique in Section 2.2.

2.1. Preliminaries

Letαandβ be two distinct points inΩ. We denote byp_α,β(ξ )the projection ofξ∈R³ on the straight line passing byαandβ andr_α,β(ξ )=dist(x,[α, β]),where “dist” denotes the Euclidean distance inR³. Form∈N^∗, we set

a^α,β_m =|α−β|

m and s_j^α,β=j a_m^α,β forj=0, . . . , m.

(5)

Forξ∈R³such thatp_α,β(ξ )∈ [α, β], we define h^α,β_m (ξ )= min

0jm

p_α,β(ξ )−α−s_j^α,β, and we set

Θ_m [α, β]

=

ξ∈R³, p_α,β(ξ )∈ [α, β]andr_α,β(ξ )a_m^α,βh^α,β_m (ξ ) .

For two points x and y in Ω, we consider the class Q(x, y) of all finite collections of segments F = ([αk, βk])ⁿ⁽_k₌^F₁⁾ such that βk=αk+1,α1=x,β_n(_F₎=y,[αk, βk] ⊂Ω andαk =βk. We define the “length” of an elementF∈Q(x, y)by

¯_w(F)=

n(F) k=1

lim inf

m→+∞

1 π

Θ_m([α_k,β_k])∩Ω

ε_α^m

k,β_k(ξ )w(ξ )dξ with

ε_α^m

k,β_k(ξ )= (h^α_m^k^,β^k(ξ ))²(a_m^α^k^,β^k)⁴ ((h^α_m^k^,β^k(ξ ))²(a_m^α^k^,β^k)⁴+r_α²

k,βk(ξ ))² .

Lemma 2.1. LetPbe a finite collection of distinct points inΩorP= ∅. For any distinct pointsx₀, y₀inΩ\Pand δ >0, there existsFδ=([α₁, β₁], . . . ,[α_n, β_n])∈Q(x₀, y₀)such that(P∪ {y₀})∩(_n₋₁

k=1[α_k, β_k] ∪ [α_n, β_n[)= ∅ and

¯_w(F)d_w(x₀, y₀)+δ.

Proof. Step 1. Assume thatwis smooth onΩ. We are going to prove that for every elementF=([α₁, β₁], . . . , [α_n, β_n])∈Q(x, y), we have

¯w(F)=

n k=1[αk,βk]

w(s)ds.

It suffices to prove that for any distinct pointsα, β∈Ω,

m→+∞lim 1 π

Θ_m([α,β])∩Ω

ε^m_k(ξ )w(ξ )dξ=

[α,β]

w(s)ds. (2.1)

Without loss of generality, we may assume that[α, β] = {(0,0)} × [0, R]and we drop the indices αandβ for simplicity. We set forj=0, . . . , m−1,

C_m^j⁺=

ξ=(ξ1, ξ2, ξ3)∈Θm [α, β] , ξ3∈

sj, sj+a_m 2

, and forj =1, . . . , m,

Cm^j⁻=

ξ=(ξ1, ξ2, ξ3)∈Θm [α, β] , ξ3∈

sj−a_m

2 , sj

.

Forξ∈Cm^j⁺∪Cm^j⁻, we havehm(ξ )= |ξ3−sj|and we get that formlarge enough,

Θm([α,β])∩Ω

ε^m_k(ξ )w(ξ )dξ=

m−1 j=0

I_m^j⁺+ m j=1

I_m^j⁻ (2.2)

(6)

with I_m^j⁺=

C_m^j⁺

|ξ₃−sj|²a⁴_mw(ξ )

(|ξ₃−s_j|²a⁴_m+r²(ξ ))²dξ forj=0, . . . , m−1,

Im^j⁻=

Cm^j⁻

|ξ₃−s_j|²a⁴_mw(ξ )

(|ξ₃−s_j|²a⁴_m+r²(ξ ))²dξ forj=1, . . . , m.

Using the change of variablez₁=_|_ξ₃^ξ₋¹_s_j_|,z₂=_|_ξ₃^ξ₋²_s_j_| andz₃=ξ₃, we derive that

I_m^j⁺=

sj+am/2 s_j B_am(0)

a_m⁴w(|z₃−s_j|z₁,|z₃−s_j|z₂, z₃) (a_m⁴ +z²₁+z²₂)² dz₁dz₂

dz₃

=

s_j+am/2 sj

w(0,0, z₃)+O(a_m)

B_am(0)

a⁴_m

(a⁴_m+z₁²+z²₂)²dz₁dz₂

dz₃

=π

s_j+a_m/2 sj

w(0,0, z3)dz3+O(a²_m).

By similar computations we get that I_m^j⁻=π

sj

sj−am/2

w(0,0, z₃)dz₃+O(a²_m).

Combining this equalities with (2.2), we obtain that

Θm([α,β])∩Ω

ε^m_k(ξ )w(ξ )dξ =π R 0

w(0,0, z₃)dz₃+O(a_m) which ends the proof of (2.1).

Step 2. We fix two distinct pointsx₀, y₀∈Ω\P. For any pointsx, yinΩ\(P∪ {y₀}), letQ(x, y)be the class of elementsF=([α₁, β₁], . . . ,[α_n, β_n])∈Q(x, y)such that

n k=1

[α_k, β_k] ⊂Ω\ P∪ {y₀} .

We consider the functionDw:Ω\(P∪ {y₀})×Ω\(P∪ {y₀})→R₊defined by Dw(x, y)= Inf

F∈Q(x,y)

(¯F).

We are going to show thatDwdefines a distance function which can be extended toΩ× Ω. Letx, y∈Ω\(P∪{y₀}) and letF=([α₁, β₁], . . . ,[α_n, β_n])be an element ofQ(x, y). Assumption (1.1) and similar computations to those in Step 1 lead to

λ n k=1

|αk−βk|¯w(F)Λ n k=1

|αk−βk|.

(7)

Taking the infimum over allF∈Q(x, y), we infer that

λ dΩ(x, y)Dw(x, y)Λ dΩ(x, y). (2.3)

From (2.3), we deduce that Dw(x, y)=0 if and only if x =y. Let us now prove that Dw is symmetric. Let x, y∈Ω\(P∪ {y₀})andδ >0 arbitrary small. By definition, we can findFδ=([α₁, β₂], . . . ,[α_n, β_n])inQ(x, y) satisfying

¯w(Fδ)Dw(x, y)+δ.

Then forF_δ=([β_n, α_n], . . . ,[β₁, α₁])∈Q(y, x), we have Dw(y, x)¯_w(Fδ)= ¯_w(Fδ)Dw(x, y)+δ.

Sinceδis arbitrary, we obtainDw(y, x)Dw(x, y)and we conclude thatDw(y, x)=Dw(x, y)inverting the roles ofxandy. The triangle inequality is immediate since the juxtaposition ofF1∈Q(x, z)withF2∈Q(z, y)is an element ofQ(x, y). HenceDwdefines a distance onΩ\(P∪ {y₀})verifying (2.3). Therefore distanceDwextends uniquely toΩ× Ω into a distance function that we still denote byDw. By continuity,Dw satisfies (2.3) for any x, y∈ Ω.

Step 3. We consider the functionζ:Ω→Rdefined by ζ (x)=Dw(x, x₀).

Note that functionζ is 1-Lipschitz with respect to distance Dw and thereforeΛ-Lipschitz with respect to the Euclidean geodesic distance onΩby (2.3). We fix an arbitrary pointz₀∈Ω\(P∪ {y₀})and someR >0 such that B_3R(z₀)⊂Ω\(P∪ {y₀}). Let(ρ_n)_n_∈Nbe a sequence of smooth mollifiers. Forn >1/R, we consider the smooth functionζ_n=ρ_n∗ζ :B_R(z₀)→R.We write

ζ_n(x)=

B_1/n

ρ_n(−z)ζ (x+z)dz

and therefore for allx, y∈B_R(z₀), ζn(x)−ζn(y)

B_1/n

ρn(−z)ζ (x+z)−ζ (y+z)dz

B_1/n

ρn(−z)Dw(x+z, y+z)dz

B_1/n

ρ_n(−z)¯_w [x+z, y+z] dz.

We remark thatΘm([x+z, y+z])=z+Θm([x, y]). Formlarge enoughz+Θm([x, y])⊂B3R(z0)and then for any vectorξ∈Θm([x, y]), we haveε^m_x₊_z,y₊_z(ξ+z)=ε_x,y^m (ξ ). Hence we obtain for allz∈B1/n(0),

¯w [x+z, y+z]

=lim inf

m→+∞

1 π

Θ_m([x,y])

ε_x,y^m (ξ )w(ξ+z)dξ.

Using Fatou’s lemma, we get that ζ_n(x)−ζ_n(y)

B1/n

ρ_n(−z)

lim inf

m→+∞

1 π

Θm([x,y])

ε^m_x,y(ξ )w(ξ+z)dξ

dz

lim inf

m→+∞

1 π

B_1/n

Θ_m([x,y])

ρn(−z)ε^m_x,y(ξ )w(ξ+z)dξdz.

(8)

For eachm∈Nsufficiently large we have 1

π

B1/n

Θm([x,y])

ρ_n(−z)ε^m_x,y(ξ )w(ξ+z)dξdz= 1 π

Θm([x,y])

ε^m_x,y(ξ )ρ_n∗w(ξ )dξ, and sinceρ_n∗wis smooth, we obtain as in Step 1,

1 π

Θm([x,y])

ε_x,y^m (ξ )ρn∗w(ξ )dξ →

[x,y]

ρn∗w(s)ds asm→ +∞. Thus for eachx, y∈B_R(z₀)we have

ζn(x)−ζn(y)

[x,y]

ρn∗w(s)ds.

Then forx∈B_R(z₀),h∈S²fixed andδ >0 small, we infer that

|ζ_n(x+δh)−ζ_n(x)|

δ 1

δ

[x,x+δh]

ρ_n∗w(s)ds −→

δ→0⁺

ρ_n∗w(x)

and we conclude, letting δ→0, that |∇ζ_n(x)·h|ρ_n∗w(x)for each x∈B_R(z₀)andh∈S² which implies that|∇ζ_n|ρ_n∗wonB_R(z₀). Since∇ζ_n→ ∇ζ andρ_n∗w→wa.e. onB_R(z₀)asn→ +∞, we deduce that

|∇ζ|wa.e. onBR(z₀). Sincez₀is arbitrary inΩ\(P∪ {y₀}), we derive

|∇ζ|w a.e. onΩ.

By Proposition 2.3. in [13], it follows that|ζ (x)−ζ (y)|d_w(x, y)for anyx, y∈ Ω and in particular, we obtain choosingy=x₀,

Dw(x, x₀)dw(x, x₀) for allx∈ Ω.

Step 4. End of the Proof. Letδ >0 be given. We choose somey˜₀∈Ω\(P∪ {y₀})such that[ ˜y₀, y₀] ⊂Ω\Pand

| ˜y₀−y₀|_3Λ^δ . By the previous step, we can find an elementF=([α₁, β₁], . . . ,[α_n, β_n])∈Q(x₀,y˜₀)verifying ¯w(F)dw(x₀,y˜₀)+δ

3.

Then we considerF=([α₁, β₁], . . . ,[α_n, β_n],[ ˜y₀, y₀])∈Q(x₀, y₀). We have ¯_w(F)¯_w(F)+Λ| ˜y₀−y₀|d_w(x₀,y˜₀)+2δ

3 d_w(x₀, y₀)+d_w(y₀,y˜₀)+2δ

3 d_w(x₀, y₀)+δ and thenFsatisfies the requirement. 2

2.2. The dipole removing technique

We first present the dipole removing technique for a simple dipole. We then treat the case of several point singularities.

Lemma 2.2. Let P and N be two distinct points in Ω and consider u∈H¹(Ω, S²)∩C¹(Ω\ {P , N}) with deg(u, P )= +1 and deg(u, N )= −1. Let F =([α₁, β₁], . . . ,[α_n, β_n]) be an element of Q(P , N ) such that N /∈_n₋₁

k=1[α_k, β_k] ∪ [α_n, β_n[. Then for anyδ >0 small enough, there exists a mapu_δ∈C¹(Ω, S ²)such that:

Ω

∇u_δ(x)²w(x)dx

Ω

∇u(x)²w(x)dx+8π¯_w(F)+δ andu_δcoincides withuoutside aδ-neighborhood of_n

k=1[α_k, β_k]included inΩ.

(9)

Proof. Let F=([α₁, β₁], . . . ,[α_n, β_n])∈Q(P , N ) such thatN /∈_n₋₁

k=1[α_k, β_k] ∪ [α_n, β_n[and fix some δ >0 small. We proceed in several steps.

Step 1. We consider a small 0< r₀< δ verifyingB_r₀(α₁)⊂Ω \ {N}. By Lemma A.1 in [1], we can findv∈ C¹(Ω\ {α₁, N}, S²)∩H¹(Ω)(recall thatα₁=P) satisfying

v(x)=







u(x) onΩ\B_r₀(α₁), R

x−α₁

|x−α₁|

onBr₀(α1), (2.4)

for some rotationRand

Ω

∇v(x)²w(x)dx

Ω

∇u(x)²w(x)dx+δ. (2.5)

LetW= {x∈R³,dist(x,[α₁, β₁]) < δ}. Forδ small enough, we haveW⊂Ω\ {N}. We setd= |α₁−β₁|. We choose normal coordinates such that α1=(0,0,0)and β1=(0,0, d). Let 0< r < ^r₂⁰. Since v is smooth on W\Br₀(α1), we can find a constantσ (r)such that|∇v|σ (r)onW\Br₀(α1). Form∈N^∗, we consider

K_m=

−a_m^α¹^,β¹ 2 ,a^α_m¹^,β¹

2 2

×

−a_m^α¹^,β¹

2 , d+a^α_m¹^,β¹ 2

.

Formlarge enough, we haveΘm([α1, β1])⊂Km⊂W. As in [1], we are going to construct in the next step a map v1∈C¹(W\ {β1}, S²)∩H¹(W )verifyingv1=vin a neighborhood of∂W and deg(v1, β1)= +1. For simplicity, we drop the indicesα₁andβ₁.

Step 2. We divideKminm+1 cubesQ^j_mdefined by Q^j_m=

−a_m 2 ,a_m

2 2

×

j−1 2

am,

j+1 2

am

forj=0, . . . , m.

Arguing as in [1], we infer from (2.4) that m

j=0

∂Q^jm

|∇v|²C r

a_m +mσ (r)²a_m²

. (2.6)

We are going to make use of a mapω_m:B_a²

m(0)⊂R²→S²defined by ω_m(x₁, x₂)= 2a_m²

a_m⁴ +x₁²+x₂²(x₁, x₂,−a²_m)+(0,0,1)

(ω_m was first introduced in [7] and we refer to the proof of Lemma 2 in [7] for its main properties). For j = 1, . . . , m, we choose an orthonormal direct basis(e^j₁, e^j₂, e^j₃)ofR³such that

v 0,0, (j−1/2)a_m

=(0,0,1) in the basis(e^j₁, e^j₂, e^j₃), and we define the mapv^m₁ :m

j=0∂Q^j_m→S²by (1) for(x₁, x₂, x₃)∈(_m

j=0∂Q^j_m)\(_m

j=1B²

a²_m(0)× {(j−1/2)a_m}), v₁^m(x₁, x₂, x₃)=v(x₁, x₂, x₃),

(2) forj=1, . . . , mand(x1, x2, x3)∈B²

a²_m/2(0)× {(j−1/2)am}, v₁^m(x₁, x₂, x₃)=ω_m

2x₁ a_m,2x₂

a_m

in the basis(e^j₁, e₂^j, e₃^j),

(10)

(3) forj =1, . . . , m, for(x₁, x₂, x₃)∈(B²

a_m²(0)\B²

a²_m/2(0))× {(j−1/2)a_m}and using cylindrical coordinates (x₁, x₂, x₃)=(ρcosθ, ρsinθ, z),

v₁^m(x₁, x₂, x₃)= A₁ρ+B₁, A₂ρ+B₂,

1−(A₁ρ+B₁)²−(A₂ρ+B₂)²

in the basis(e^j₁, e^j₂, e^j₃), whereA₁, A₂, B₁, B₂are determined to makev₁^mcontinuous. More precisely, if we write v=v₁e₁^j+v₂e₂^j+v₃e₃^jthen











a²_mA_i(θ )+B_i(θ )=v_i a²_mcosθ, a²_msinθ, (j−1/2)a_m

, i=1,2, a_m²

2 A₁(θ )+B₁(θ )= 2a³_m

a⁴_m+a_m² cosθ, a_m²

2 A₂(θ )+B₂(θ )= 2a³_m a⁴_m+a_m² sinθ.

The mapv₁^msatisfies by constructionv^m₁ =von∂K_m. Moreover, it follows exactly as in the proof of Lemma 2 in [1] that deg(v^m₁, ∂Q^j_m)=0 forj=0, . . . , m−1 and deg(v^m₁, ∂Q^m_m)= +1. Then we extendv^m₁ on each cubeQ^j_m by setting

v₁^m(x)=v₁^m

am(x−bj) 2x−b_j_∞+b_j

onQ^j_mforj=0, . . . , m,

whereb_j =(0,0, s_j)is the barycenter ofQ^j_mandx−b_j∞=max(|x₁|,|x₂|,|x₃−s_j|). We easily check that v^m₁ ∈H¹(K_m, S²),v₁^m=von∂K_m,v₁^mis continuous except at the pointsb_j and Lipschitz continuous outside any small neighborhood of the pointsb_j. We also get that

deg(v₁^m, b_m)= +1 and deg(v₁^m, b_j)=0 forj=0, . . . , m−1. (2.7) We remark that if we set

D_m^j =B_a²₂

m/2(0)×

(j−1/2)a_m

∪B_a²₂

m/2(0)×

(j+1/2)a_m

forj=1, . . . , m−1, D_m⁰ =B_a²₂

m/2(0)× {1/2a_m} and D^m_m=B_a²₂

m/2(0)×

(m−1/2)a_m , then we have

m j=0

x∈Q^j_m, a_m(x−b_j)

2x−bj∞+b_j∈D_m^j ifx=b_jorx=b_j otherwise

=Θ_m [α₁, β₁]

and ifx∈Q^j_m∩Θ_m([α₁, β₁])for somej ∈ {0, . . . , m}then h_m(x)= |x₃−s_j| = x−b_j∞ and r(x)=

x₁²+x₂². (2.8)

Some classical computations (see [1] and [7]) lead to, forj=0, . . . , m,

(∂Q^j_m)\D_m^j

|∇v^m₁|²

∂Q^j_m

|∇v|²+O(a_m²)

and therefore

Q^jm\Θm([α1,β1])

∇v^m₁(x)²w(x)dxC1Λam

∂Q^jm

|∇v|²+C2Λa³_m.

The relaxed energy for <span class="mathjax-formula">${S}^{2}$</span>-valued maps and measurable weights

The relaxed energy for S 2 -valued maps and measurable weights L’énergie relaxée des applications à valeurs dans S 2

et poids mesurables

Vincent Millot

The relaxed energy for S ² -valued maps and measurable weights L’énergie relaxée des applications à valeurs dans S ²