2 Basic properties of the regularizable minimiz- ers

Asymptotic behavior of regularizable minimizers of a Ginzburg-Landau functional in higher

dimensions ^∗

Yutian Lei

Abstract

We study the asymptotic behavior of the regularizable minimizers of a Ginzburg-Landau type functional. We also dicuss the location of the zeroes of the minimizers.

1 Introduction

LetG⊂Rⁿ (n≥2) be a bounded and simply connected domain with smooth boundary ∂G. Let g be a smooth map from ∂G into Sⁿ⁻¹ satisfying d = deg(g, ∂G)6= 0. Consider the Ginzburg-Landau-type functional

Eε(u, G) = 1 p Z

G

|∇u|^p+ 1 4ε^p

Z

G

(1− |u|²)², (p >1)

with a small parameter ε > 0. It is known that this functional achieves its minimum on

Wp={v∈W^1,p(G,Rⁿ) :v|∂G=g}

at a functionu_ε. We are concerned with the asymptotic behavior ofu_εand the location of the zeroes ofu_εas ε→0.

The functionalEε(u, G) was introduced in the study of the Ginzburg-Landau vortices by F. Bethue, H. Brezis and F. Helein [1] in the casep=n= 2. Similar models are also used in many other theories of phase transition. The minimizer uεofEε(u, G) represents a complex order parameter. The zeroes ofuεand the module |uε| both have physics senses, for example, in superconductivity |uε|² is proportional to the density of supercoducting electrons, and the zeroes ofuε

are the vortices, which were introduced in the type-II superconductors.

In the case 1< p < n, it is easily seen that W_g^1,p(G, Sⁿ⁻¹)6=∅. It is not difficult to prove that the existence of solutionupfor the minimization problem

min{ Z

G

|∇u|^p :u∈W_g^1,p(G, Sⁿ⁻¹)}

∗Mathematics Subject Classifications: 35J70.

Key words: Ginzburg-Landau functional, module and zeroes of regularizable minimizers.

2001 Southwest Texas State University.c

Submitted December 13, 2000. Published February 23, 2001.

1

by taking the minimizing sequence. This solution is called a map of the least p-energy with boundary valueg. Using the variational methods, we can proved that the solutionup is also p-harmonic map onG with the boundary data g, namely, it is a weak solution of the following equation

−div(|∇u|^p−2∇u) =u|∇u|^p.

Asε→0, there exists a subsequenceuε_k ofuε, the minimizer ofEε(u, G), such that

uε_k→up, in W^1,p(G,Rⁿ).

In the case p > n, W_g^1,p(G, Sⁿ⁻¹) = ∅. Thus there is no map of least p-energy on G with the boundary value g. It seems to be very difficult to study the convergence for minimizers of E_ε(u, G) in W_p. Some results on the asymptotic behavior of the radial minimizers ofEε(u, G) were presented in [7].

Whenp=n, this problem was introduced in [1] (the open problem 17). M.

C. Hong studied the asymptotic behavior for the regularizable minimizers of Eε(u, G) inWn [6]. He proved that there exist{a1, a2, . . . , aJ} ⊂G,J ∈N and a subsequenceuε_k of the regularizable minimizersuε such that

uε_k

→w un, inW_loc^1,n(G\ {a1, a2, . . . , aJ},Rⁿ) (1.1) asεk →0, whereun is an n-harmonic map.

In this paper we shall discuss the asymptotic behavior for the regularizable minimizers of Eε(u, G) on Wn in the case p = n. Without loss of generality, we may assumed >0. Recalling a minimizer ofEε(u, G) on Wn be called the regularizable minimizer, if it is the limit of the minimizer of the regularized functional

E_ε^τ(u, G) = 1 p Z

G

(|∇u|²+τ)^p/2+ 1 4ε^p

Z

G

(1− |u|²)², (τ∈(0,1)) onW_n in W^1,p. It is not difficult to prove that the regularizable minimizer is also a minimizer ofE_ε(u, G). In order to find the zeroes of the minimizers, we should first locate the singularities of the n-harmonic mapu_n.

Theorem 1.1 If aj ∈ G, j = 1,2, . . . , J are the singularities of n-harmonic map u_n, then J =d, the degree deg(u_n, a_j) = 1, and {a_j}^dj=1 ⊂G. Moreover, for everyj, there exists at least one zero of the regularizable minimizeru_εnear toa_j.

Because the module of the minimizer has the physics sense, we have also studied its asymptotic behavior.

Theorem 1.2 Let uε be a regularizable minimizer of Eε(u, G),ρ=|uε|, then there exists a constant C independent ofε such that

Z

G

|∇ρ|ⁿ≤C, and 1 εⁿ

Z

G

(1−ρ²)≤C(1 +|lnε|).

For any given η >0, denote Gη=G\ ∪^dj=1B(aj, η), then asε→0, 1

εⁿ Z

G_η

(1−ρ²)²→0, ρ→1, inC_loc(G_η, R).

At last, we develop the conclusion of (1.1) into following

Theorem 1.3 There exists a subsequenceuε_k of uε such that as ε→0, uε_k→un, in W_loc^1,n(G\ ∪^dj=1{aj},Rⁿ).

We shall prove Theorems 1.2 and 1.3 in§5 and§7 respectively, and the proof of Theorem 1.1 will be given in §6.

2 Basic properties of the regularizable minimiz- ers

First we recall the minimizer of the regularized functional E_ε^τ(u, G) = 1

n Z

G

(|∇u|²+τ)^n/2+ 1 4εⁿ

Z

G

(1− |u|²)², τ ∈(0,1) on Wn, denoted by u^τ_ε. As τ → 0, there exists a subsequence u^τ_ε^k of u^τ_ε such that

τlim_k→0u^τ_ε^k =uε, inW^1,n(G,Rⁿ), (2.1) and the limitu_εis one minimizer ofE_ε(u, G) onW_n, which is named the regu- larizable minimizer. It is not difficult to prove thatu^τ_ε solves the problem

−div[(|∇u|²+τ)^(n−2)/2∇u] = 1

εⁿu(1− |u|²), onG, (2.2) u|∂G=g

and satisfies the maximum principle: |u^τ_ε| ≤1 on G. Moreover

Proposition 2.1 (Theorem 2.2 in [6]) For any δ > 0, there exists a con- stant C independent ofε such that

limτ→0|∇u^τ_ε| ≤Cε⁻¹, onG^δε, (2.3) where G^δε ={x∈G: dist(x, ∂G)≥δε}.

In this section we shall present some basic properties of the regularizable minimizeruε. Clearly it is a weak solution of the equation

−div(|∇u|ⁿ⁻²∇u) = 1

εⁿu(1− |u|²), onG, (2.4) and it is known that|uε| ≤1 a.e. onG[6]. We also have

Proposition 2.2 For any δ > 0, there exists a constant C independent of ε such that

k∇u_εkL^∞(B(x,δε/8,Rⁿ)≤Cε⁻¹, ifx∈G^δε.

Proof. Let y=xε⁻¹ in (2.4) and denote v(y) =u(x),Gε={y=xε⁻¹ :x∈ G}, G^δ = {y ∈ Gε : dist(y, ∂Gε) > δ}. Since that u is a weak solution of (2.4), we have

Z

Gε

|∇v|ⁿ⁻²∇v∇φ= Z

Gε

v(1− |v|²)φ, φ∈W₀^1,n(Gε,Rⁿ).

Takingφ=vζⁿ, ζ ∈C₀^∞(Gε, R), we obtain Z

Gε

|∇v|ⁿζⁿ≤n Z

Gε

|∇v|ⁿ⁻¹ζⁿ⁻¹|∇ζ||v|+ Z

Gε

|v|²(1− |v|²)ζⁿ.

Setting y ∈ G^δ, B(y, δ/2) ⊂ Gε, and ζ = 1 in B(y, δ/4), ζ = 0 in Gε\ B(y, δ/2),|∇ζ| ≤C(δ), we have

Z

B(y,δ/2)

|∇v|ⁿζⁿ ≤C(δ) Z

B(y,δ/2)

|∇v|ⁿ⁻¹ζⁿ⁻¹+C(δ).

Using Holder inequality we can derive R

B(y,δ/4)|∇v|ⁿ ≤C(δ). Combining this with the theorem of [9] yields

k∇vkⁿL^∞(B(y,δ/8))≤C(δ) Z

B(y,δ/4)

(1 +|∇v|)ⁿ ≤C(δ) which implies

k∇ukL^∞(B(x,εδ/8))≤C(δ)ε⁻¹.

Proposition 2.3 (Lemma 2.1 in [6]) There exists a constantCindependent of εsuch that forε∈(0,1),

E_ε(u_ε, G)≤d(n−1)^n/2

n |Sⁿ⁻¹||lnε|+C. (2.5) Proposition 2.4 There exists a constant C independent ofε such that

1 εⁿ

Z

G

(1− |ue|²)²≤C. (2.6)

Proof. By (3.6) in [6], Z

G

|∇uε|ⁿ≥d(n−1)^n/2|Sⁿ⁻¹||lnε| −C.

Applying Proposition 2.3 we may obtain (2.6).

3 A class of bad balls

Fixρ >0. For the regularizable minimizeruε, from Theorem 2.2 in [6] we know

|uε| ≥ 1

2, onG\G^ρε, (3.1)

where G^ρε ={x ∈G : dist(x, ∂G) ≥ρε}. Thus there exists no zero of u_ε on G\G^ρε.

Proposition 3.1 Let u_ε be a regularizable minimizer of E_ε(u, G), There exist positive constants λ, µ which are independent ofε∈(0,1)such that if

1 εⁿ

Z

G^ρε∩B^2lε

(1− |uε|²)²≤µ, (3.2) where B^2lε is some ball of radius2lεwithl≥λ, then

|uε| ≥ 1

2, ∀x∈G^ρε∩B^lε. (3.3)

Proof. First it is known that there exists a constantβ >0 such that for any x∈G^ρεand 0< r≤1,

|G^ρε∩B(x, r)| ≥βrⁿ. Next we take

λ= min( 1 4C,1

8ρ), µ= βλⁿ 16 where Cis the constant in Proposition 2.2.

Suppose that there is a pointx0∈G^ρε∩B^lε such that|uε(x0)|<1/2, then applying Proposition 2.2 we have

|uε(x)−uε(x0)| ≤Cε⁻¹|x−x0|=1

4, x∈B(x0, λε)∩G^ρε. Hence

(1− |u_ε(x)|²)²> 1

16, ∀x∈B(x₀, λε)∩G^ρε, Z

B(x₀,λε)∩G^ρε

(1− |u_ε|²)²> 1

16|G^ρε∩B(x₀, λε)| ≥β 1

16(λε)ⁿ =µεⁿ. (3.4) Since x0 ∈ B^lε∩G^ρε, we have (B(x0, λε)∩G^ρε) ⊂ (B^2lε∩G^ρε), thus (3.4) implies

Z

B^2lε∩G^ρε

(1− |uε|²)²> µεⁿ which contradicts (3.2) and thus the proposition is proved.

To find the zeroes of the regularizable minimizer uε based on Proposition 3.1, we may take (3.2) as the ruler to distinguish the ball of radius λε which contain the zeroes.

Letλ, µ be constants in Proposition 3.1. If 1

εⁿ Z

G^ρε∩B(x^ε,2λε)

(1− |u_ε|²)²≤µ,

thenB(x^ε, λε) is called good ball. OtherwiseB(x^ε, λε) is called bad ball. From Proposition 3.1 we are led to

|u_ε| ≥ 1

2, onG^ρε\ ∪x^ε∈ΛB(x^ε, λε), (3.5) where Λ is the set of the centres of all bad balls. (3.5) and (3.1) imply that the zeroes ofuεare contained in these bad balls.

Now suppose that{B(x^ε_i, λε), i∈I} is a family of balls satisfying (i)x^ε_i ∈G^ρε, i∈I

(ii)G^ρε⊂ ∪i∈IB(x^ε_i, λε) (iii)

B(x^ε_i, λε/4)∩B(x^ε_j, λε/4) =∅, i6=j . (3.6) LetJ_ε={i∈I:B(x^ε_i, λε) is a bad ball}.

Proposition 3.2 There exists a positive integer N which is independent of ε such that the number of bad balls cardJ_ε≤N.

Proof. Since (3.6) implies that every point in G^ρε can be covered by finite, say m (independent of ε) balls, from (2.6) and the definition of bad balls,we have

µεⁿcardJε ≤ X

i∈Jε

Z

B(x^ε_i,2λε)∩G^ρε

(1− |uε|²)²

≤ m Z

∪i∈JεB(x^ε_i,2λε)∩G^ρε

(1− |uε|²)²

≤ m Z

G

(1− |u_ε|²)²≤mCεⁿ and hence cardJ_ε≤ ^mC_µ ≤N.

Similar to the argument of Theorem IV.1 in [1], we have

Proposition 3.3 There exist a subsetJ ⊂Jε and a constant h≥λsuch that

∪i∈J_εB(x^ε_i, λε)⊂ ∪i∈JB(x^ε_j, hε),

|x^ε_i−x^ε_j|>8hε, i, j∈J, i6=j. (3.7)

Proof. If there are two pointsx1, x2 such that (3.7) is not true withh=λ, we take h1 = 9λ and J1 =Jε\ {1}. In this case, if (3.7) holds we are done.

Otherwise we continue to choose a pair pointsx3, x4which does not satisfy (3.7) and takeh2= 9h1andJ2=Jε\ {1,3}. After at mostN steps we may conclude this proposition.

Applying Proposition 3.3 we may modify the family of bad balls such that the new one, denoted by {B(x^ε_i, hε) :i∈J}, satisfies

∪i∈J_εB(x^ε_i, λε)⊂ ∪i∈JB(x^ε_i, hε),

λ≤h; cardJ ≤cardJ_ε, (3.8)

|x^ε_i −x^ε_j|>8hε, i, j∈J, i6=j.

The last condition implies that every two balls in the new family do not intersect.

Asε→0, there exist a subsequencex^ε_i^k ofx^ε_i and a_i ∈Gsuch that x^ε_i^k→ai, i= 1,2, . . . , N1= cardJ.

Perhaps there may be at least two subsequences converge to the same point, we denote by

a1, a2, . . . , aN2, N2≤N1

the collection of distinct points in{ai}^N1¹.

To proveaj∈∂G, it is convenient to enlarge a littleG. AssumeG⁰⊂Rⁿ is a bounded, simply connected domain with smooth boundary such that G⊂G⁰, and take a smooth map ¯g : (G⁰ \G) → Sⁿ⁻¹ such that ¯g = g on ∂G. We extend the definition domain of every element in {u:G→Rⁿ :u|∂G =g} to G⁰ such thatu=gonG⁰\G. In particular, the regularizable minimizeruεcan be defined onG⁰.

Fix a small constantσ >0 such that

B(a_j, σ)⊂G⁰, j= 1,2, . . . , N₂; 4σ <|aj−ai|, i6=j; 4σ <dist(G, ∂G⁰).

Writing Λ_j={i∈J:x^ε_i^k→a_j}, j= 1,2, . . . , N₂, we have

∪i∈Λ_jB(x^ε_i^k, hεk)⊂B(aj, σ), j= 1,2, . . . , N2

∪j∈JB(x^ε_j^k, hεk)⊂ ∪^Nj=1² B(aj, σ/4) B(x^ε_i^k, hε_k)∩B(x^ε_j^k, hε_k) =∅, i, j∈J, i6=j

as long asεk is small enough. Letuεis the regularizable minimizer ofEε(u, G) and denote d^k_i =deg(uε_k, ∂B(x^ε_i^k, hεk)), l_j^k=deg(uε_k, ∂B(aj, σ)), thus

l^k_j = X

i∈Λj

d^k_i, d=

N2

X

j=1

l^k_j. (3.9)

To prove that the degrees d^k_i and l^k_j are independent of εk, we recall a proposition stated in [6] (Lemma 3.3) or [2] (Theorem 8.2).

Proposition 3.4 Let φ:Sⁿ⁻¹→Sⁿ⁻¹ be aC⁰-map withdegφ=d. Then Z

Sⁿ⁻¹

|∇τφ|ⁿ⁻¹dx≥ |d|(n−1)^(n−1)/2|Sⁿ⁻¹|.

Proposition 3.5 There exists a constant C which is independent of ε_k such that

|d^k_i| ≤C, i∈J; |l^k_j| ≤C, j= 1,2, . . . , N₂.

Proof. Sinceu=u_εis a weak solution of (2.4), applying the theory of the local regularity in [9], we know u∈C(∂B(x^ε_i^k, hεk)). Since (3.5) implies |u| ≥ 1/2 on∂B(x^ε_i^k, hεk), thusφ= ^u

|u| ∈C(∂B(x^ε_i^k, hεk), Sⁿ⁻¹). From Proposition 3.4, we have

|d^k_i| ≤ |Sⁿ⁻¹|⁻¹(n−1)^(1−n)/2 Z

∂B(x^εk_i ,hε_k)

|( u

|u|)_τ|ⁿ⁻¹. Since|u| ≥ ¹₂ onG⁰\G^ρε, there is no zero ofu_εin it. Thus

deg(uε_k, ∂B(x^ε_i^k, hεk)) = deg(uε_k, ∂(B(x^ε_i^k, hεk)∩G^ρεk)) and

|d^k_i| ≤ |Sⁿ⁻¹|⁻¹(n−1)^(1−n)/2 Z

∂[B(x^εk_i ,hε_k)∩G^ρε]

|( u

|u|)_τ|ⁿ⁻¹. (3.10) Substituting (2.3) and the fact|uε_k| ≥¹₂ on∂[B(x^ε_i^k, hεk)∩G^ρε] into (3.10), we obtain

|d^k_i| ≤Cε¹_k⁻ⁿ|Sⁿ⁻¹|⁻¹(n−1)^(1−n)/2(hεk)ⁿ⁻¹≤C,

where C is a constant which is independent ofεk. Combining this with (3.9) we can complete the proof of the proposition.

Proposition 3.5 implies that there exist a number k_j which is independent ofε_k and a subsequence ofl^k_j denoted itself such that

l^k_j →kj, as k→ ∞.

Since l^k_j, k_j ∈N,{l_j^k} must be constant sequence for any fixed j, namely l^k_j = kj. The same reason shows d^k_i can be writen as di which is also a number independent ofεk later.

4 An estimate for the lower bound

Write Ω⁰ =G⁰\ ∪^Nj=1² B(aj, σ). Fixingj∈ {1,2, . . . , N2} and takingi0∈Λj, we havexi₀ →aj as ε→0. Thus

∪i∈ΛjB(x^ε_i, hε)⊂B(x_i0, σ/4)⊂B(a_j, σ) (4.1) holds withεsmall enough.

Denote Ωj =B(aj, σ)\ ∪i∈Λ_jB(x^ε_i, hε),Ωjσ =B(xi₀, σ/4)\ ∪i∈Λ_jB(x^ε_i, hε).

To estimate the lower bound of k∇uεkLⁿ(Ωj), the following proposition is nec- essary that was given by Theorem 3.9 in [6].

Proposition 4.1 Let As,t(xi) = (B(xi, s)\B(xi, t))∩G with ε≤t < s≤R.

Assume thatu∈W_g^1,n(G,Rⁿ)and¹₂ ≤ |u| ≤1onAs,t(xi). If there is a constant C such that

1 εⁿ

Z

A_s,t(x_i)

(1− |u|²)²≤C.

Then for ε < ε0 there holds Z

A_s,t(x_i)

|∇u|ⁿ≥ |di|^n/(n−1)(n−1)^n/2|Sⁿ⁻¹|lns t −C,

where C is a constant which is independent of ε and di is the degree of u on each ∂(B(x_i, r)∩G), t≤r≤s.

Proposition 4.2 AssumeCardΛj =N. Then Z

Ω_j

|∇u_ε|ⁿ≥ Z

Ω_j,σ

|∇u_ε|ⁿ≥(n−1)^n/2|Sⁿ⁻¹||k_j|lnσ

ε −C (4.2) where C is a constant which is independent ofε.

Proof. We give the proof following that in [6] (see Theorem 3.10), and the idea comes from [8]. Supposex1, x2, . . . , xN converge toaj, anddi,R(i= 1,2, . . . , N) is the degree of uε around ∂B(xi, R). Let R^σ_ε denote the set of all numbers R ∈ [ε, σ] such that∂B(xi, R)∩B(xj, ε) = ∅ for all i 6= j and such that for some collection JR ⊂ {1,2, . . . , N}, satisfying JR ⊂JR⁰ ifR⁰ ≤R, the family {B(xi, R)}i∈JR is disjoint and

∪^Ni=1B(xi, ε)⊂ ∪i∈J_R0B(xi, R⁰)⊂ ∪i∈J_RB(xi, R), R⁰≤R.

Note that R^σ_ε is the union of closed intervals [R^l₀,R^l],1 ≤l ≤L, whose right endpoints correspond to a number R=R^l such that∂B(x_i, R)∩B(x_j, R)6=∅ for some pairi6=j∈JR and whose left endpoints correspond to a number R^l₀ such thatB(xi,R^l−1)\ ∪j∈J₀B(xj, R^l₀)6=∅fori∈J_Rl

0. JR=J^lis a constant for R ∈[R^l₀,R^l] andJ^l+1 ⊂J^l, J^l+16=J^l. ThusL≤N. Moreover, there exists a constant M =M(h)>0 such that

R^l₀≤M ε, R^L ≥σ/M, R^l+1₀ ≤M R^l (4.3) for alll= 1,2, . . . , L−1. Finally, observe that for allR∈R^σ_ε andJ ∈JR,

|kj|=| X

i∈JR

di,R| ≤ X

i∈JR

|di,R|^n/(n−1). (4.4) Applying (4.3)(4.4) and proposition 4.1 we have

Z

Ω_j,σ

|∇uε|ⁿ ≥

L

X

l=1

X

i∈J^l

| Z

A_{Rl ,Rl}

0

(x_i)

∇uε|ⁿ

≥

L

X

l=1

X

i∈J^l

|Sⁿ⁻¹|(n−1)^n/2|d_i,Rl|ln(R^l/R^l₀)−C

≥ |Sⁿ⁻¹|(n−1)^n/2|kj|X

l

(lnR^l−lnR^l₀)−C

≥ (n−1)^n/2|Sⁿ⁻¹||kj|lnσ ε −C.

This and (4.1) imply that (4.2) holds.

Remark In fact the following results Z

Ω_j

|∇ uε

|u_ε||ⁿ≥(n−1)^n/2|Sⁿ⁻¹||k_j|^n/(n−1)lnσ ε,

and Z

Ω_j

(1− |uε|ⁿ)|∇ u_ε

|uε||ⁿ≤C

had been presented in the proof of Theorem 3.9 in [6], where C which is inde- pendent ofε. Noticing

Z

Ω_j

|u_ε|ⁿ|∇u_ε

|uε||ⁿ = Z

Ω_j

|∇ u_ε

|uε||ⁿ− Z

Ω_j

(1− |u_ε|ⁿ)|∇ u_ε

|uε||ⁿ, we have

Z

Ω_j

|uε|ⁿ|∇ u_ε

|uε||ⁿ≥(n−1)^n/2|kj|^n/(n−1)|Sⁿ⁻¹|lnσ ε −C.

Theorem 4.3 There exists a constant C which is independent ofε, σ ∈(0,1) such that

Z

∪^Nj=1²Ω_j

|∇uε|ⁿ≥(n−1)^n/2|Sⁿ⁻¹|dlnσ

ε −C, (4.5)

1 n

Z

G_σ

|∇uε|ⁿ+ 1 4εⁿ

Z

G

(1− |uε|²)²≤ 1

n(n−1)^n/2|Sⁿ⁻¹|dln 1

σ+C (4.6) whereGσ=G\ ∪^Nj=1² B(aj, σ).

Proof. From (4.2) and Proposition 2.3 we have (n−1)^n/2|Sⁿ⁻¹|(

N₂

X

j=1

|k_j|) lnσ

ε ≤(n−1)^n/2|Sⁿ⁻¹|dln1 ε +C or (PN₂

j=1|kj| −d) ln¹_ε ≤C. It is seen asεsmall enough

N₂

X

j=1

|kj| ≤d=

N₂

X

j=1

kj

which implies

kj ≥0. (4.7)

This and (3.9) imply

N₂

X

j=1

|kj|=

N₂

X

j=1

kj =d. (4.8)

Substituting (4.8) into (4.2) yields (4.5), and (4.6) may be concluded from (4.5) and Proposition 2.3.

From (4.6) and the fact |u_ε| ≤ 1 a.e. on G, we may conclude that there exists a subsequenceu_εk ofu_ε such that

uε_k

→w u_∗, W^1,n(Gσ,Rⁿ) (4.9) as εk→0. Compare (4.9) with (1.1) we knownu_∗=un onGσ, and

{a_j}^N_j=1² ={a_j}^Jj=1. (4.10) These points were called the singularities ofun.

To show these singularitiesaj∈∂G, the following conclussion is necessary.

Proposition 4.4 Assumea∈∂Gandσ∈(0, R)with a small constant R. If u∈W^1,n(A_R,σ(a), Sⁿ⁻¹)∩C⁰, u=g

on (G⁰\G)∩B(a, R)anddeg(u, ∂B(a, R)) = 1, then there exists a constantC which is independent of σsuch that

Z

A_R,σ(a)

|∇u|ⁿ ≥2¹ⁿ(n−1)^n/2|Sⁿ⁻¹|ln1

σ−C . (4.11)

Proof. Similar to the proof of Lemma VI.1 in [1], we may writeGas the half space

{(x1, x2, . . . , xn) :xn>0} locally and aas 0 by a conformal change.

Denote St =∂B(0, t), t∈(σ, R). Noticing that g is smooth on G⁰\G, we have

sup

G⁰\G

|g_τ| ≤C₁. Takingtsufficiently small such that

t≤(n−1)^1/2(2ⁿ⁻¹−1)^1/(n−1) 2C1

, then

Z

S_t⁻

|g¯_τ|ⁿ⁻¹≤ |S_t⁻|C₁ⁿ⁻¹≤ |Sⁿ⁻¹|tⁿ⁻¹C₁ⁿ⁻¹≤(n−1)^(n−1)/2|Sⁿ⁻¹|(1−2¹⁻ⁿ) (4.12)

with R <1 small enough, where S_t⁻ =St∩ {xn <0}. On the other hand we can be led to

(n−1)^(n−1)/2|Sⁿ⁻¹| ≤ Z

St

|u_τ|ⁿ⁻¹= Z

S_t⁺

|u_τ|ⁿ⁻¹+ Z

S_t⁻

|g¯_τ|ⁿ⁻¹ from Proposition 3.4. HereS_t⁺=St\S⁻_t . Combining this with (4.12) yields

Z

S⁺_t

|uτ|ⁿ ≥ |S_t⁺|^−1/(n−1)( Z

S⁺_t

|uτ|ⁿ⁻¹)^n/(n−1) (4.1)

≥ 2ⁿ¹|Sⁿ⁻¹|(n−1)^n/2t⁻¹. (4.2) Integrating this over (σ, R), we obtain

Z

A_R,σ

|∇u|ⁿ≥2¹ⁿ|Sⁿ⁻¹|(n−1)^n/2lnR σ

which implies (4.11). To provek_j = 1 for anyj, we supposeR >2σ is a small constant such that

B(aj, R)⊂G⁰; B(aj, R)∩B(ai, R) =∅, i6=j. (4.13) Denote Π ={v∈W^1,n(Ω⁰, Sⁿ⁻¹)∩C⁰: deg(v, ∂B(a_j, r)) =k_j, r ∈(σ, R), j = 1,2, . . . , N2}.

Proposition 4.5 For anyv∈Π, ifkj ≥0, j= 1,2, . . . , N2, then there exists a constant C=C(R)which is independent of σsuch that

Z

Ω⁰

|∇v|ⁿ≥(n−1)^n/2|Sⁿ⁻¹|(

N₂

X

j=1

k

n n−1

j ) ln1

σ−C. (4.14)

Proof. WriteA_R,σ(a_j) =B(a_j, R)\B(a_j, σ), thus∪^N_j=1² A_R,σ(a_j)⊂Ω⁰. From Proposition 3.4 we have

kj=|kj| ≤ (n−1)^(1−n)/2|Sⁿ⁻¹|⁻¹ Z

Sⁿ⁻¹

|vτ|ⁿ⁻¹

≤ (n−1)^(1−n)/2|Sⁿ⁻¹|^(n−1)/n( Z

Sⁿ⁻¹

|v_τ|ⁿ)^(n−1)/n namely

Z

Sⁿ⁻¹

|v_τ|ⁿ≥(n−1)^n/2|Sⁿ⁻¹|k^n/(n_j ⁻¹⁾. On the other hand, we may obtain

Z

Ω⁰

|∇v|ⁿ ≥

N₂

X

j=1

Z

AR,σ(aj)

|∇v|ⁿ

≥

N2

X

j=1

Z R

σ

Z

Sⁿ⁻¹

r⁻ⁿ|∇τv|ⁿrⁿ⁻¹dζdr

≥ (n−1)^n/2|Sⁿ⁻¹|

N₂

X

j=1

k_j^n/(n−1) Z R

σ

r⁻¹dr

= (n−1)^n/2|Sⁿ⁻¹|(

N2

X

j=1

k^n/(n_j ⁻¹⁾) lnR σ which implies (4.14).

5 The proof of Theorem 1.2

Let uεbe a regularizable minimizer ofEε(u, G). Proposition 2.4 has given one estimate of convergence rate of |uε|. Moreover, we also have

Theorem 5.1 There exists a constantCwhich is independent ofε∈(0,1)such

that 1

εⁿ Z

G

(1− |u_ε|²)≤C(1 + ln1

ε). (5.1)

Proof. The minimizer u =u^τ_ε of the regularized functional E_ε^τ(u, G) solves (2.2). Taking the inner product of the both sides of (2.2) withuand integrating overGwe have

1 εⁿ

Z

G

|u|²(1− |u|²) = − Z

G

div(v^(n−2)/2∇u)u

= Z

G

v^(n−2)/2|∇u|²− Z

∂G

v^(n−2)/2uun (5.2)

≤ Z

G

v^(n−2)/2|∇u|²+C Z

∂G

v^n/2+C

where n denotes the unit outward normal to ∂G and u_n the derivative with respect to n.

To estimateR

∂Gv^n/2, we choose a smooth vector fieldν such thatν|∂G=n.

Multiplying (2.2) by (ν· ∇u) and integrating overG, we obtain 1

εⁿ Z

G

u(1− |u|²)(ν· ∇u) = − Z

G

div(v^(n−2)/2∇u)(ν· ∇u)

= Z

G

v^(n−2)/2∇u·(ν· ∇u)− Z

∂G

v^(n−2)/2|un|². Combining this with

1 εⁿ

Z

G

u(1− |u|²)(ν· ∇u) = 1 2εⁿ

Z

G

(1− |u|²)(ν· ∇(|u|²))

= − 1

4εⁿ Z

G

(1− |u|²)²divν

and Z

G

v^(n−2)/2∇u· ∇(ν· ∇u)

= Z

G

v^(n−2)/2|∇u|²divν+1 n

Z

G

ν· ∇(v^n/2)

= Z

G

v^(n−2)/2|∇u|²divν+1 n

Z

∂G

v^n/2− 1 n

Z

G

v^n/2divν we obtain

Z

∂G

v^(n−2)/2|un|²≤ C 4εⁿ

Z

G

(1− |u|²)²+C Z

G

v^n/2+ 1 n

Z

∂G

v^n/2. Thus

Z

∂G

v^n/2 = Z

∂G

v^(n−2)/2(|u_n|²+|g_t|²+τ)

≤ C Z

∂G

v^(n−2)/2+1 n

Z

∂G

v^n/2+CE_ε^τ(u^τ_ε, G).

Substituting this into (5.2) yields 1

εⁿ Z

G

|u|²(1− |u|²)≤CE_ε^τ(u^τ_ε, G).

Letτ→0, applying (2.1) and Proposition 2.3 we have 1

εⁿ Z

G

|uε|²(1− |uε|²)≤CEε(uε, G)≤C(1 +|lnε|) which and (2.6) imply (5.1).

Theorem 5.2 Denoteρ=|u_ε|. There exists a constantCwhich is independent of ε∈(0,1)such that

k∇ρkLⁿ(G)≤C. (5.3)

Proof. Denote u=uε. From the Remark in§4 we know Z

Ωj

|u|ⁿ|∇ u

|u||ⁿdx≥(n−1)^n/2|kj|ⁿ⁻¹ⁿ |Sⁿ⁻¹|lnσ ε −C.

Thus we may modify (4.5) as Z

∪^N_j=1²Ω_j

ρⁿ|∇ u

|u||ⁿ≥(n−1)^n/2|Sⁿ⁻¹|dlnσ ε −C.

Combining this with Z

∪^Nj=1² Ω_j

|∇u|ⁿ ≥ Z

∪^Nj=1²Ω_j

ρⁿ|∇ u

|u||ⁿ+ Z

∪^Nj=1² Ω_j

|∇ρ|ⁿ−C

and Proposition 2.3, we derive Z

∪^Nj=1²Ω_j

|∇ρ|ⁿ≤C. (5.4)

On the other hand, from (2.1) and Proposition 2.1 we are led to Z

G^ρε∩B(xi,hε)

|∇uε|ⁿ= lim

τ_k→0

Z

G^ρε∩B(xi,hε)

|∇uετ_k

|ⁿ ≤C(λε)ⁿ(C/ε)ⁿ≤C, fori∈Λj. Summarizing fori and using (5.4) we can obtain (5.3).

Theorem 5.3 For theσ >0 in Theorem 4.4, then as ε→0, 1

εⁿ Z

G_3σ

(1−ρ²)²→0, (5.5)

where G3σ=G\ ∪^Nj=1² B(aj,3σ).

Proof. The regularizable minimizeru_εsatisfies Z

Gσ

|∇u|ⁿ⁻²∇u∇φ= 1 εⁿ

Z

Gσ

uφ(1− |u|²), (5.6) where φ ∈ W₀^1,n(G_σ,Rⁿ) since u_ε is a weak solution of (2.4). Denoting u = u^τ_ε = ρw, ρ =|u|, w = ^u

|u| in G_σ and taking φ =ρwζ, ζ ∈ W₀^1,n(G_σ,Rⁿ), we have

Z

G_σ

|∇u|ⁿ⁻²(w∇ρ+ρ∇w)(ρζ∇w+ρw∇ζ+wζ∇ρ) = 1 εⁿ

Z

G_σ

ρ²ζ(1−ρ²). (5.7) Substituting 2w∇w=∇(|w|²) = 0 into (5.7), we obtain

Z

Gσ

|∇u|ⁿ⁻²(ρ∇ρ∇ζ+|∇u|²ζ) = 1 εⁿ

Z

Gσ

ρ²ζ(1−ρ²). (5.8) SetS={x∈Gσ:ρ(x)>1−ε^β}for some fixedβ∈(0, n/2) andρ= max(ρ,1− ε^β), thusρ=ρonS. In (5.8) takingζ= (1−ρ)ψ, whereψ∈C^∞(Gσ, R), ψ= 0 onGσ\G2σ,0< ψ <1 onG2σ\G3σ, ψ= 1 onG3σ, we have

Z

G_σ

|∇u|ⁿ⁻²ρ∇ρ· ∇ρψ¯ + 1 εⁿ

Z

G_σ

l²(1−ρ²)(1−ρ)ψ¯ (5.9)

= Z

Gσ

|∇u|ⁿ⁻²ρ∇ρ∇ψ(1−ρ) +¯ Z

Gσ

|∇u|ⁿψ(1−ρ) Noticing 1/2≤l≤1 inGσ and applying (4.6) we obtain

1 εⁿ

Z

G3σ

(1−ρ)(1−ρ²) + Z

S∩G3σ

|∇u|ⁿ⁻²|∇ρ|²≤Cε^β. (5.10)

On the other hand, (2.6) implies ε^2β|G_σ\S| ≤

Z

Gσ\S

(1−l²)²≤Cεⁿ,

namely|G_σ\S| ≤Cε^n−2β. Then there exists a small constantε₀>0 such that G_3σ ⊂S∪E

asε∈(0, ε0) whereE is a set, the measure of which converges to zero. Thus lim

ε→0

Z

G_3σ

(1−ρ²)(1−ρ) = lim

ε→0

Z

G_3σ

(1 +ρ)(1−ρ)². By (5.10),

εlim→0

1 εⁿ

Z

G_3σ

(1 +ρ)²(1−ρ)²

≤ lim

ε→0

2 εⁿ

Z

G3σ

(1−ρ)(1−ρ²) = 0 This is our conclusion.

Theorem 5.4 Assume B(x,2σ)⊂G_σ satisfies 1

εⁿ Z

B(x,σ)

(1− |uε|²)²→0, asε→0, (5.11) then|uε| →1 inC(B(x, σ), R).

Proof. SinceB(x,2σ)⊂Gσ, there existsε0sufficiently small so thatB(x, σ)⊂ G^2δε0. We always assumeε < ε0. Forx0∈B(x, σ), setα=|uε(x0)|. Proposi- tion 2.2 implies

|u_ε(x)−u_ε(x₀)|< Cε⁻¹τ ε, ifx∈B(x₀, τ ε),

whereτ= (1−α)(N C)⁻¹, Cis the constant in Proposition 2.2 andN is a large number such thatτ < δ. ThusB(x0, τ ε)⊂B(x, σ) and

|uε(x)| ≤α+Cτ, ifx∈B(x0, τ ε), Z

B(x0,τ ε)

(1− |uε(x)|²)²≥(1−1/N)²(1−α)ⁿ⁺²πεⁿ(N C)⁻ⁿ.

Combining this with (5.11) we obtain (1−α)ⁿ⁺² =o(1) as ε→0. Thus it is not difficult to complete the proof of Theorem.

6 The proof of Theorem 1.1

It is known that the singularities ofun are inGfrom the discussion in§3. Since deg(g, ∂G) > 0, we can see that the zeroes of u_ε are also in G . Moreover, the zeroes are contained in finite bad balls, i.e. B(x^ε_i, hε), i ∈ J. As ε → 0, B(x^ε_i, hε)→a_j, i∈Λ_j. This implies that the zeroes ofu_εdistribute near these singularities of u_n as ε→0. Thus it is necessary to describe these singularities {a_j}, j= 1,2, . . . , N₂.

Proposition 6.1 kj = deg(un, aj).

Proof. Denote Ω⁰=G⁰\ ∪^Nj=1² B(a_j, σ). Combining (4.6) and Z

G⁰\G

|∇u_ε|ⁿ= Z

G⁰\G

|∇g¯|ⁿ≤C, we have

Z

Ω⁰

|∇uε|ⁿ ≤C+ (n−1)^n/2|Sⁿ⁻¹|d|lnσ|, (6.1) whereC is a constant which is independent ofε. ForRin (4.13), from (6.1) we

have Z

AR,σ(aj)

|∇uε|ⁿ ≤C.

Then we know that there exists a constantr∈(σ, R) such that Z

∂B(a_j,r)

|∇uε|ⁿ≤C(r)

by using integral mean value theorem. Thus there exists a subsequence uε_k of uεsuch that

uε_k →un, inC(∂B(aj, r)) as ε_k→0, which implies

k_j= deg(u_ε, ∂B(a_j, σ)) = deg(u_n, a_j).

Proposition 6.2 k_j = 0or k_j= 1.

Proof. From the regularity results on n-harmonic maps (see [3][5] or [9]), we knowun∈C⁰(Gσ,Rⁿ). Set

w=

g¯ onG⁰\G, u_n onG_σ, thenw∈Π. Using Proposition 4.5 and (4.7) we have

Z

Ω⁰

|∇w|ⁿ≥(n−1)^n/2|Sⁿ⁻¹|(

N₂

X

j=1

k

n n−1

j ) ln1

σ−C(R). (6.2)

On the other hand, (6.1) and (4.9) imply uε_k

→w w, inW^1,n(Ω⁰,Rⁿ).

Noting this and the weak lower semicontinuity of R

Ω⁰|∇u|ⁿ, applying (6.1) we have

Z

Ω⁰

|∇w|ⁿ ≤lim_εk→0 Z

Ω⁰

|∇uε_k|ⁿ≤(n−1)^n/2|Sⁿ⁻¹|dln 1

σ+C. (6.3) Combining this with (6.2), we obtain

(

N2

X

j=1

k

n n−1

j −d) ln1

σ ≤C or

N2

X

j=1

k

n n−1

j ≤d=

N2

X

j=1

k_j

forσsmall enough. Thus (k_j^1/(n−1)−1)kj ≤0 which implies that the Proposition holds.

Proposition 6.3 kj >0,j= 1,2, . . . , N2.

Proof. Supposek1= 0 andk2, k3, . . . , kN₂>0. Similar to the proof of Theo- rem 4.3 we have

Z

∪^Nj=2²Ωj

|∇uε|ⁿ≥(n−1)^n/2|Sⁿ⁻¹|dlnσ ε −C.

By this we can rewrite (4.6) as Z

G\∪^Nj=2² B(aj,σ)

|∇u_ε|ⁿ+ 1 4εⁿ

Z

G

(1− |u_ε|²)²≤C(σ).

Thus similar to the proof of Theorem 5.3 we may modify (5.5) as 1

εⁿ Z

G\∪^Nj=2²B(a_j,3σ)

(1− |u_ε|²)²→0 (6.4) asε→0. Noticing

G∩B(a1, σ)⊂G∩B(a1, R)⊂G\ ∪^Nj=2² B(aj, R)⊂G\ ∪^Nj=2² B(aj,3σ) we have

1 εⁿ

Z

G∩B(a1,σ)

(1− |uε|²)²→0. (6.5) On the other hand, the definition ofa1implies that there exists at least one bad ballB(x^ε₀, hε) such that

G∩B(x^ε₀, hε)⊂G∩B(a1, σ).

Applying the definition of bad ball we obtain 1

εⁿ Z

G∩B(a1,σ)

(1− |uε|²)²≥ 1 εⁿ

Z

G∩B(x^ε₀,hε)

(1− |uε|²)²≥µ >0 which is contrary to (6.5). This contradiction showsk1>0.