On the limit <span class="mathjax-formula">$ p\to \infty $</span> of global minimizers for a p-Ginzburg–Landau-type energy

Ann. I. H. Poincaré – AN 30 (2013) 1159–1174

www.elsevier.com/locate/anihpc

On the limit p → ∞ of global minimizers for a p-Ginzburg–Landau-type energy

Yaniv Almog

, Leonid Berlyand

, Dmitry Golovaty

, Itai Shafrir

aDepartment of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA bDepartment of Mathematics, Pennsylvania State University, University Park, PA 16802, USA cDepartment of Theoretical and Applied Mathematics, The University of Akron, Akron, OH 44325, USA

dDepartment of Mathematics, Technion – Israel Institute of Technology, 32000 Haifa, Israel Received 18 July 2012; received in revised form 27 November 2012; accepted 7 December 2012

Available online 21 January 2013

Abstract

We study the limitp→ ∞of global minimizers for ap-Ginzburg–Landau-type energy E_p(u)=

R²

|∇u|^p+1 2

1− |u|²2 .

The minimization is carried over maps onR²that vanish at the origin and are of degree one at infinity. We prove locally uniform convergence of the minimizers onR²and obtain an explicit formula for the limit onB(0,√

2). Some generalizations to dimension N3 are presented as well.

©2013 Elsevier Masson SAS. All rights reserved.

1. Introduction

For anyd∈Z,N2 andp > Nconsider the class of maps Ep^d=

u∈W_loc^1,p

R^N,R^N

: Ep(u) <∞, deg(u)=d , where

E_p(u)=

R^N

|∇u|^p+1 2

1− |u|²2

.

By deg(u) we mean the degree of u “at infinity”, which is properly defined since by Morrey’s inequality (cf.[4, Theorem 9.12]), for any mapu∈W_loc^1,p(R^N,R^N)with

R^N|∇u|^p<∞we have u∈C_loc^α

R²,R²

, whereα=1−N/p

* Corresponding author.

E-mail address:almog@math.lsu.edu(Y. Almog).

0294-1449/$ – see front matter ©2013 Elsevier Masson SAS. All rights reserved.

http://dx.doi.org/10.1016/j.anihpc.2012.12.013

(except, perhaps, for a set of measure zero inR²) and

u(x)−u(y)C_p,N∇uL^p(R^N)|x−y|^α, ∀x, y∈R^N. (1) In fact, according to the proof given in[4], one can select

C_p,N= 2^2−N/p

1−N/p. (2)

It then easily follows (see[1]for the caseN=2; the proof for any integer value ofN >2 is identical) that

|xlim|→∞u(x)=1. (3)

Consequently,uhas a well-defined degree, deg(u), equal to the degree of theS^N−1-valued map_|^u_u| on any large circle {|z| =R},R1.

In what follows, we assume thatN=2 and, whenever appropriate, interpretR²-valued maps as complex-valued functions of the variablez=x+iy. We will return to the caseN3 at the end of the Introduction and present some partial results for this case (Section4).

For anyd∈Z, let I_p(d)=inf

E_p(u): u∈Ep^d

. (4) It has been established in[1] that Ip(1)is attained for each p >2 andN =2. Denote by up a global minimizer of Ep inEp¹. It is clear thatEp is invariant with respect to translations and rotations. However, it is still unknown whether uniqueness of the minimizer u_p, modulo the above symmetries, is guaranteed. Such a uniqueness result would imply that, up to a translation and a rotation,u_p must take the formf (r)e^iθ (withr= |x|). Note that radial symmetry of a nontriviallocal minimizerin the casep=2 was established by Mironescu in[7](with a contribution from Sandier[8]). One way of inquiring whether the global minimizeru_pis radially symmetric or not forp >2, is by looking at the limiting behavior of{u_p}p>2asp→ ∞, which is the focus of the present contribution. We have already studied in [2]the behavior of minimizers in the class of radially symmetric functions whenpis large and, in addition, showed their local stability for 2< p4. The results presented in this work seem to support the radial symmetry conjecture (as in the casep=2[7]); indeed, in the limitp→ ∞, we obtain the same asymptotic behavior foru_pas in the case of radially symmetric minimizers[2].

In view of the translational and rotational invariance properties ofE_p, we may assume for eachp >2 that

u_p(0)=0 and u_p(1)∈ [0,∞). (5)

Our first main result is the following

Theorem 1.For eachp >2, letu_pdenote a minimizer ofE_pinEp¹satisfying(5). Then, for a sequencep_n→ ∞, we haveu_pn→u_∞inC_loc(R²)and weakly in _p>1W_loc^1,p(R²,R²), whereu_∞satisfies

u_∞(z)=^√z₂ onB(0,√

2)= {|z|<√ 2}, u_∞(z)=1 onR²\B(0,√

2).

(6) Furthermore, the convergence|u_pn| → |u_∞|is uniform onR².

Theorem1 fails to identify the values inS¹ that the mapu_∞ assumes onR²\B(0,√

2). A natural conjecture appears to be thatu_∞(z)=_|^z_z| onR²\B(0,√

2), i.e., thatu_∞=F where

F (z)=

⎧⎨

⎩

√z

2 onB(0,√ 2),

|zz| onR²\B(0,√

2). (7)

For simplicity, whenever appropriate, we will use the abbreviated notation u_p foru_pn. Our second main result establishes explicit estimates for the rate of convergence ofu_ptou_∞inside the discB(0,√

2).

Theorem 2.Under the assumptions of Theorem1, for everyβ <1anda <√

2, there existsC_β,a>0such that for all p >2,

u_p−u_∞L^∞(B(0,a))C_β,a

p^β/2. (8)

Finally we consider the minimization ofEpin dimensions higher than 2. Although it is presently unknown whether Ip(1)is attained for everyp > N3, by using the same technique as in the proof of Theorem1we can show that the minimizer ofE_pexists for sufficiently large values ofp:

Theorem 3.For everyN3there existspNsuch that for everyp > pN the minimum valueIp(1)ofEp is attained inE_p¹by someu_p∈W_loc^1,p(R^N,R^N).

In view of Theorem3it makes sense to investigate the asymptotic behavior of the set of minimizers{up}p>2asp tends to infinity for everyN3. This is presented in the following

Theorem 4.For eachp > p_N, letu_p denote a minimizer ofE_p inEp¹ satisfyingu_p(0)=0. Then, for a sequence pn→ ∞, we have

u_pn→u_∞inC_loc R^N

and weakly in

p>1

W_loc^1,p

R^N,R^N

, (9)

whereu_∞satisfies

u_∞(x)=^√U_N^x onB(0,√ N ),

|u_∞(x)| =1 onR^N\B(0,√ N ),

(10) for some orthogonalN×N matrixUwithdet(U)=1. We also have

∇u_∞L^∞(R^N)=1 (11)

and the convergence|u_p| → |u_∞|is uniform onR^N.

Remark 1.1.We may alternatively state that (subsequences of) minimizers ofE_poverE_p¹satisfyingu(0)=0 converge to a minimizer for the following problem:

inf

R^N

1− |u|²2

: u∈W^1,∞

R^N,R^N

, u(0)=0, ∇u_∞1

. (12)

The latter result can, most probably, be appropriately formulated in terms ofΓ-convergence. Theorem4shows that the minimizers of (12) are given by the set of maps inW^1,∞(R^N,R^N)satisfying (10)–(11). The infinite size of this set is the source of our difficulty in identifying the limit mapu_∞outside the ballB(0,√

N ). To confirm the natural conjecture thatu_∞(x)=^U_|x^x_| for|x|>√

N, a more delicate analysis of the energiesEp(up)or of the Euler–Lagrange equation satisfied byupis required. In fact, our present arguments can be used to prove the same convergence result as in Theorem4 not only for the minimizers{up}, but also for a sequence of “almost minimizers”{vp}, satisfying E_p(v_p)I_p(1)+o(1)asp→ ∞.

2. Proof of Theorem1

We first recall the upper-bound for the energy that was proved in[2]using the test functionU_p(re^iθ)=f_p(r)e^iθ with

f_p(r)=

⎧⎪

⎨

⎪⎩

√1

2(1−^ln_p^p)r, r <

√2 1−^ln_p^p,

1, r ^√2

1−^lnp^p

.

Lemma 2.1.We have I_p(1)π

3 +Clnp

p , ∀p >3. (13)

Remark 2.1.From (13) we clearly obtain that

R²

|∇u_p|^pC, ∀p >3, (14)

whereCis independent ofp. While this estimate is sufficient for our purpose, it should be noted that one can derive a more precise estimate

R²

|∇up|^p=2

pIp(1)C p,

via a Pohozaev-type identity (see[1, Lemma 4.1]).

Our next lemma provides a key estimate that will lead to a lower-bound forIp(1).

Lemma 2.2.Letρ∈(0,1)be a regular value ofu_p(which by Sard’s lemma holds for almost everyρ)and set A_ρ=

z∈R²: u_p(z)< ρ

. (15)

Then, for any componentV_ρofA_ρwithdeg(u, ∂Vρ)=d, we have for largep

V_ρ

1− |up|²2

|d|

4π ρ⁴

2 −ρ⁶ 3

+o(1)

, (16)

whereo(1)denotes a quantity that tends to zero aspgoes to infinity, uniformly forρ∈(0,1).

Proof. Sinceρis a regular value ofup, we can conclude from (3) that∂Vρis a finite union of closed and simpleC¹- curves, and hence deg(u, ∂Vρ)is well-defined. Since the image ofV_ρ byu_ρ covers the discB(0, ρ)(algebraically) d times, it follows by Hölder’s inequality that

π|d|ρ²=

Vρ

(u_p)_x×(u_p)_y 1

2

Vρ

|∇u_p|²1 2μ(V_ρ)

p−2 p

Vρ

|∇u_p|^p _p²

, (17)

whereμdenotes the Lebesgue measure inR², which, in turn, yields μ(Vρ) (2π|d|ρ²)

p−2p

(

Vρ|∇u_p|^p)^p−22

. (18)

From (18) and (14), we get

Vρ

1− |u_p|²2

= 1 (1−ρ²)²

μ

1− |u_p|²2

> t

∩V_ρ dt

= ρ 0

4r 1−r²

μ(A_r∩V_ρ) dr ρ 0

4r

1−r² (2π|d|r²)^pp−2 (

V_r|∇u_p|^p)^p−22 dr

|d| ρ

0

4r

1−r² 2π r²

dr+o(1)

= |d|

4π ρ⁴

2 −ρ⁶ 3

+o(1)

. 2 (19)

Corollary 2.1.There existρ₀∈(³₄,1), p0andR₀such that for allp > p₀the setA_ρ0has a componentV_ρ0⊂B(0, R0) for whichdeg(u, ∂Vρ0)=1and|u_p|¹₂onR²\V_ρ0.

Proof. Note that by (2) one can select uniformly boundedC_p,2in (1) forp3. This fact, together with (14) implies equicontinuity of the maps{u_p}p3onR². Therefore, there existsλ >0 such that

u_p(z0)1

2 ⇒ u_p(z)3

4 onB(z0, λ) ⇒

B(z0,λ)

1− |u_p|²2

ν:=π λ² 7

16 2

. (20)

Fixρ0∈(³₄,1)such that 4π

ρ₀⁴ 2 −ρ₀⁶

3

>max π

3,2π 3 −ν

. (21)

LetV_ρ0 be a component ofA_ρ0 with deg(up, ∂V_ρ0)=0 (we may assume w.l.o.g. thatρ₀is a regular value ofu_p).

By (13), (16) and (21), it follows that there can be only one such component whenpis sufficiently large (and thus deg(up, ∂V_ρ0)=1). Moreover, by (20) and (21), on any other component of A_ρ0 (if there is one) we must have

|up|>¹₂.

It remains necessary to show thatVρ₀ is embedded in a sufficiently large disc. Similarly to (20), there existsλ0>0 such that

u_p(z₀)ρ₀ ⇒ u_p(z)1+ρ₀

2 onB(z₀, λ₀)

⇒

B(z₀,λ₀)

1− |u_p|²2

ν₀:=π λ²₀

1−

1+ρ₀ 2

22

. (22)

SinceVρ₀ is connected and 0∈Vρ₀, the set{|z|: z∈Vρ₀}is the interval[0, R)for some positiveR. For any integer k for which 2kλ0R there exists a set of points {zj}^k_j⁻₌¹₀⊂Vρ₀ with|zj| =2j λ0. By (22) and (13) we have for sufficiently largepthat

kν₀

k−1

j=0

B(zj,λ0)

1− |u_p|²2

< c₀:=2π 3 +1.

It follows thatRis bounded from above byR0:=2λ0(^c_ν⁰

0+1). 2

In order to complete the proof of Theorem1we need to establish the convergence of{u_pn}^∞_n=1tou_∞and to identify the limit. We begin with the following lemma

Lemma 2.3.For a sequencep_n→ ∞we have

nlim→∞u_pn=u_∞inC_loc R²

and weakly in

p>1

W_loc^1,p R²,R²

. (23)

Furthermore, the limit mapu_∞is a degree-one map inW^1,∞(R²,R²)satisfying also(5)and

∇u_∞∞1. (24)

Proof. Fix anyq >3. Sinceu_pL^∞1 (see[1]), we have by (13) on each discB(0, m), m1, that u_pW^1,q(B(0,m))C_m, p > q.

It follows that for allm1, there exists a sequencep_n↑ ∞, such that{u_pn}converges weakly inW^1,q(B(0, m))to a limitu_∞. By Morrey’s theorem, the convergence holds inC(B(0, m))as well. Since the latter is true for everym1

and everyq >3, we may apply a diagonal subsequence argument to find a subsequence satisfying (23). The fact that u_∞has degree one too follows from (23) and Corollary2.1.

Finally, in order to prove (24), it suffices to note that for any discB⊂R²,λ >1 andq >1, we have by (14) and the weak lower semicontinuity of theL^q-norm,

λ^qμ

|∇u_∞|> λ

∩B

B

|∇u_∞|^qlim inf

p→∞

B

|∇u_p|^qlim inf

p→∞ μ(B)^1−q/p

B

|∇u_p|^p q/p

μ(B). (25)

Lettingqtend to∞in (25) yieldsμ({|∇u_∞|> λ} ∩B)=0. The conclusion (24) follows since the discB andλ >1 are arbitrary. 2

A similar argument to the one used in the proof of Lemma2.2yields Proposition 1.

plim→∞

1 2

R²

1− |u_p|²2

= lim

p→∞I_p(1)=π 3 =1

2

R²

1− |F|²2

,

whereF is as defined in(7).

Proof. As in (18) we have μ(A_ρ) (2πρ²)

p p−2

(

Aρ|∇u_p|^p)^p−22

. (26)

Therefore,

R²

1− |up|²2

= 1 0

μ

1− |up|²2

> t dt

= 1 0

4ρ 1−ρ²

μ(A_ρ) dρ 1 0

4ρ

1−ρ² (2πρ²)^pp−2 (

A_ρ|∇u_p|^p)^p2−2

dρ. (27)

Since

Aρ|∇u_p|^pI_p(1)C, taking the limit inferior of both sides of (27) yields, with the aid of (7) lim inf

p→∞

R²

1− |up|²2

1 0

4ρ

1−ρ² lim inf

p→∞

2πρ²_p−^p₂ dρ

= 1 0

4ρ 1−ρ²

μ

|F|< ρ dρ=

R²

1− |F|²2

=2π

3 , (28)

and the proposition follows by combining (28) with (13). 2

Remark 2.2.In fact, for anydwe have limp→∞I_p(d)=^|d₃^|π (see Proposition2in Section4).

We can now complete the proof of Theorem1.

Proof of Theorem1. For eachρ∈(0,1], letD_ρ= {z∈R²: |u_∞(z)|< ρ}. Using arguments similar to those used to establish Proposition1, we obtain

R²

1− |u_∞|²2

= 1 0

μ

1− |u_∞|²2

> t dt=

1 0

4ρ 1−ρ²

μ(D_ρ) dρ. (29)

Since deg(u_∞)=1 by Lemma2.3, using (24) yields πρ²

Dρ

(u_∞)_x×(u_∞)_y

Dρ

(u_∞)_x×(u_∞)_y1 2

Dρ

|∇u_∞|²1

2μ(D_ρ). (30)

From (29)–(30) it follows that

R²

1− |u_∞|²2

1

0

8πρ 1−ρ²

ρ²dρ=2π

3 . (31)

On the other hand, by Lemma2.3and Proposition1, for everyR >0

B(0,R)

1− |u_∞|²2

= lim

n→∞

B(0,R)

1− |up_n|²2

2π

3 , which together with (31) implies that

R²

1− |u_∞|²2

=2π

3 . (32)

Therefore, for anyρ∈(0,1), pointwise equalities between the integrands in (30) must hold almost everywhere inDρ. It follows that

(u_∞)x⊥(u_∞)y, (u_∞)x=(u_∞)y and (u_∞)x²+(u_∞)y²=1, sign

(u_∞)x×(u_∞)y

≡σ ∈ {−1,1},

(33) a.e. in D₁. From (33) we conclude that u_∞ is a conformal map a.e. inD₁ (it cannot be anti-conformal because deg(u_∞)=1) with|∇u_∞| ≡1. Hence,u_∞must be of the formu_∞(z)=az+bwith|a| =^√1

2. Sinceu_∞satisfies (5), we finally conclude that (6) holds.

Finally, to prove that|u_p| → |u_∞|uniformly onR²assume, on the contrary, that for someρ₀<1 there exists a sequence{z_n}^∞_n=1with|z_n| → ∞such that|u_pn(z_n)|ρ₀for alln. But then using (22) we are led immediately to a contradiction with Proposition1since we have already established that

nlim→∞

R²

1− |u_pn|²2

= lim

n→∞

B(0,√ 2)

1− |u_pn|²2

=2π

3 . 2

3. Proof of Theorem2 Let

∂

∂z

def= 1 2

∂

∂x −i ∂

∂y

; ∂

∂z¯

def= 1 2

∂

∂x +i ∂

∂y

.

We begin with a simple lemma that establishes the existence of an approximate holomorphic map for a given mapu such that theL²-norm of^∂u_∂z¯ is “small”. To this end we introduce some additional notation. For a functionf∈L¹(Ω) we denote byf_Ω its average value overΩ, i.e.,

fΩ= 1 μ(Ω)

Ω

f.

We further set∇_⊥u=(u_y,−u_x).

Lemma 3.1.LetΩbe a bounded, simply connected domain inR²with∂Ω∈C¹. Letu=u_r+iu_i∈H¹(Ω,C)satisfy

Ω

|∇u+i∇_⊥u|²², (34)

for some >0. Then, there existsvwhich is holomorphic inΩand such thatvΩ=uΩ,

Ω

∇(u−v)²4² (35)

and

Ω

|∇u|²=

Ω

|∇v|²+

Ω

∇(u−v)². (36) Proof. Consider the Hilbert spaceH= {U∈H¹(Ω,C): U_Ω=0}with the normU²_H=

Ω|∇U|²and its closed subspaceK= {V ∈H: V is holomorphic inΩ}. Letv=V +u_Ω whereV ∈K is the nearest point projection of u−u_Ω∈HonK. Clearlyvsatisfies (36). To prove (35), it is sufficient, in view of the definition ofv, to construct a single functionv˜∈H¹(Ω,C), which is holomorphic inΩ, and satisfies

Ω

∇(u− ˜v)²4². (37) Setv˜= ˜v_r+iv˜_iwherev˜_r∈H₀¹(Ω,C)+u_r is harmonic andv˜_iis the conjugate harmonic function tov˜_r satisfying (v˜_i)_Ω=(u_i)_Ω. Letφ∈C₀^∞(Ω,C). Clearly,

Ω

∇ ¯φ· ∇⊥w=0, ∀w∈H¹(Ω,C), (38)

and sincev˜is harmonic, we have

Ω

∇ ¯φ· ∇ ˜v=0. (39)

By density ofC₀^∞(Ω,C)inH₀¹(Ω,C), (38)–(39) hold for everyφ∈H₀¹(Ω,C). In particular, employing the identity

∇ ˜v+i∇_⊥v˜=0, (40)

and using (38) we obtain forφ=u_r− ˜v_r that ∇(u_r− ˜v_r)²

2=

Ω

∇(u_r− ˜v_r)· ∇(u− ˜v)=

Ω

∇(u_r− ˜v_r)·

∇(u− ˜v)+i∇_⊥(u− ˜v)

=

Ω

∇(u_r− ˜v_r)·(∇u+i∇⊥u)∇(u_r− ˜v_r)

2∇u+i∇⊥u

2. (41)

Hence, by (34) and (41), ∇(u_r− ˜v_r)

2. (42)

Setw=u− ˜v. By (34) and (40)

∇w+i∇_⊥w2. (43)

However, asw_r is real we have by (42)

∇w_r+i∇_⊥w_r2=√

2∇w_r2√

2. (44)

Since

∇w+i∇_⊥w= ∇wr+i∇_⊥wr+i(∇wi+i∇_⊥wi), we get from (43)–(44) that

∇wi2= 1

√2∇wi+i∇⊥wi2 1

√2

∇w+i∇⊥w2+ ∇wr+i∇⊥wr2

1+ 1

√2

,

which together with (42) clearly implies (37) 2

By Poincaré inequality and (35) we immediately deduce:

Corollary 3.1.Letvbe given by Lemma3.1. Then,

u−vH¹(Ω)C, (45)

whereCdepends only onΩ.

Lemma 3.2.Letf be holomorphic inΩ⊂R². Suppose that for every discB(x0, s)⊂Ωwe have

B(x0,s)

|f|²−1

, (46)

for some >0. Then,

f²L^∞(Ω_s)1+ μ(B(x₀, s)), where

Ωs=

x∈Ωd(x, ∂Ω) > s .

Proof. Asf is holomorphic,|f|²is subharmonic. By the mean value principle we obtain for anyx₀∈Ω_s f (x₀)² 1

μ(B(x₀, s))

B(x₀,s)

|f|²=1+ 1 μ(B(x₀, s))

B(x₀,s)

|f|²−1

, (47)

from which the lemma easily follows. 2

Lemma 3.3.Letf be holomorphic inB_R=B(0, R)⊂R². Suppose that

BR

1− |f|²

, (48)

for some >0. Suppose further that

f²L^∞(B_R)1+. (49)

Then, there existα∈ [−π, π )andC >0, depending only onR, such that f (x)−e^iαC

d_x², x∈B_R, (50)

whered_x=R− |x|. Proof. By (48)–(49),

BR

|f|²−1−=

BR

1− |f|²

+π R²C,

hence,

B_R

|f|²−1C (51) (we denote byCandcdifferent constants, depending onR only). Since the function||f|²−1|is subharmonic, we deduce from (51) that for everyx∈B_R,

f (x)²−1 1 π d_x²

B(x,dx)

|f|²−1c

d_x². (52)

It follows in particular that f (x)²1

2, |x|R−√

2c. (53)

In B(0, R−√

2c )we may write thenf =e^U+iV, whereV is the conjugate harmonic function ofU that satisfies V (0)∈ [−π, π ). By (52) we have

U (x)C

d_x², |x|R−√

2c. (54)

From (54) we get an interior estimate for the derivatives ofU(see (2.31) in[5]):

∇U (x)C

d_x³, |x|R−√

4c. (55)

Note that by the Cauchy–Riemann equations, (55) holds forV as well, i.e., ∇V (x)C

d_x³, |x|R−√

4c. (56)

For anyx∈B(0, R−√

4c )\ {0}we obtain, using (56), the estimate V (x)−V (0)

R d_x

∇V

(R−s) x

|x|

dsC R d_x

ds s³ C

d_x². (57)

Therefore, settingα=V (0)and using (54) and (57), we obtain for everyx∈B(0, R−√

4c )that f (x)−e^iαf (x)−e^{iV (x)}+e^{iV (x)}−e^{iV (0)}e^{U (x)}−1+V (x)−V (0)C

d_x². Forx∈B_R\B(0, R−√

4c ), i.e., whend_x√

4c, we have clearly|f (x)−e^iα|2+, so choosingCbig enough yields (50) for allx∈B_R. 2

LetA_ρ be defined in (15). The following lemma lists some of its properties.

Lemma 3.4.There existp₀>2andC >0such that for allp > p₀andρ >¹₂we have μ(A_ρ)2πρ²

1−C

p

, (58a)

1 0

ρ

1−ρ²μ(A_ρ)−2πρ²dρClnp

p . (58b)

Proof. The estimate (58a) follows directly from (26) and (14). Since by (58a) μ(A_ρ)μ(A_ρ)−2πρ²+2πρ²−C

p, we obtain using (27) that

Ip(1) 1 0

2ρ 1−ρ²

μ(Aρ) dρ 1 0

2ρ

1−ρ²μ(A_ρ)−2πρ²dρ+π 3 −C

p.

Combining the above with (13) yields (58b). 2

Lemma 3.5.Letlnp/pδ_p<1/4. There exists1−2δp< ρ <1−δ_p, such that for allp > p₀

Aρ

|∇u_p+i∇_⊥u_p|²Cδ_p⁻²lnp

p . (59)

Proof. By (58b) there exists 1−2δp< ρ <1−δpsuch that μ(A_ρ)−2πρ²Cδ⁻_p²lnp

p . (60)

Applying (60) yields 1

4

Aρ

|∇u_p+i∇⊥u_p|²=

Aρ

1

2|∇u_p|²−(u_p)_x×(u_p)_y

=

A_ρ

1

2|∇up|²−πρ²1 2

A_ρ

|∇up|^p 2/p

μ(Aρ)^1−2/p−πρ²

1

2

1+C p

2πρ²+Cδ_p⁻²lnp p

1−2/p

−πρ² C δ_p²

lnp

p . 2 (61)

Proof of Theorem2. Setη=^√2₁₀^−a and then b_j=a+j η, j=1, . . . ,9.

Letρ be given by Lemma3.5for δp=η/√

2, so thatρ∈(b8/√ 2, b9/√

2). We can also assume without loss of generality thatρis a regular value for|u_p|. By Theorem1we have for sufficiently largep,

B(0, b8)⊂A_ρ⊂B(0, b9). (62)

By (62) and Lemma3.5we have

B(0,b₈)

|∇up+i∇⊥up|²

A_ρ

|∇up+i∇⊥up|² C (a−√

2)² lnp

p =Ca

lnp p .

Applying Corollary 3.1 yields the existence of a holomorphic function v_p in B(0, b8) such that (v_p)_B(0,b8) = (u_p)_B(0,b8)and such that (36) holds withu=u_p, v=v_pand

up−vp²_H1(B(0,b8))Ca

lnp

p . (63)

We denotew_p(z)=√

2v_p(z)(wherev_p =^∂v_∂z^p is the derivative of the holomorphic mapv_p) and note that|w_p(z)| =

|∇v_p(z)|. Asais kept fixed, we suppress in the sequel the dependence of the constants ona.

For any ballB⊂B(0, b8)we apply the same estimates as in (17),

B

|∇u_p|²−1

B(0,b₈)

|∇u_p|^p 2/p

μ(B)^1−2/p−μ(B)(1+C/p)

μ(B)1−2/p

−μ(B)C p.

Combining the above with (36) yields

B

|w_p|²−1

=

B

|∇v_p|²−1

B

|∇u_p|²−1C

p, ∀B⊂B(0, b8).

By Lemma3.2it then follows that w_p²_L∞(B(0,b7))1+C₁

p . (64)

Next, we apply Lemma3.5again, this time withδ_p=3η/√

2, to find a correspondingρ˜∈(b₄/√ 2, b7/√

2). For plarge we haveB(0, b4)⊂A_ρ˜⊂B(0, b7). Arguing as in (17) we obtain, using (60),

A_ρ˜

|∇u_p|²−12

A_ρ˜

(u_p)_x×(u_p)_y−μ(A_ρ˜)2πρ˜²−μ(A_ρ˜)−Clnp p .

By (36), once again, we have that

A_ρ˜

|wp|²−1

−Clnp

p . (65)

Next, we apply the same argument as the one used in the beginning of the proof of Lemma3.3to obtain, using (64) and (65),

A_ρ˜

|w_p|²−1−C₁ p

=

A_ρ˜

1−C₁

p − |w_p|²

Clnp p .

Hence, also

B(0,b₄)

|wp|²−1

A_ρ˜

|wp|²−1Clnp

p . (66)

We can now use (64) and (66) and apply Lemma3.3to obtain the existence ofα_p∈ [−π, π )such that w_p(z)−e^iαpClnp

p , z∈B(0, a). (67)

Consequently, there exists a constantγ_psuch that √

2vp(z)−e^iαpz−γpClnp

p , z∈B(0, a). (68)

Set

U=u_p−v_p.

For everyq >2 we have forp > q, by (63), (68), and the fact that|up|1, U^q_Lq(B(0,a))U^q_L⁻∞²(B(0,a))U²_L2(B(0,a))C

lnp p

.

Furthermore, by Hölder’s inequality, (67), (63) and (13) we have that

∇U^q_Lq(B(0,a))∇U²

p−q p−2

L²(B(0,a))∇U^p

q−2 p−2

L^p(B(0,a))C lnp

p ^p_p⁻₋^q₂

.

Consequently, for each fixedq >2 we have UW^1,q(B(0,a))C

lnp p

^p−q

q(p−2)

. (69)

By Sobolev embedding the bound in (69) holds also forUL^∞(B(0,a)) and, in particular, we get that for every 0<

β <1,

UL^∞(B(0,a))C_βp^−β/2. (70)

Combining (70) and (68) we obtain that √

2up(z)−e^iαpz−γ_pC_βp^−β/2, z∈B(0, a).

Asu_p(0)=0, it immediately follows that|γ_p|C_βp^−β/2, and hence √

2up(z)−e^iαpzC_βp^−β/2, z∈B(0, a). (71)

Substitutingz=1 into (71) we obtain using (5) that|α_p|C_βp^−β/2and (8) follows. 2 4. The problem in dimensionN3

This section is mainly devoted to the proofs of Theorem 3and Theorem 4. We begin with the computation of limp→∞Ip(d). Denote byωN the volume of the unit ball inR^N. It turns out that the constant

τ_N:= 4ωN

(N+2)(N+4)N^N/2 (72)

generalizes the constant^π₃ in (13) for dimensions higher thanN=2.

Proposition 2.We have

plim→∞I_p(d)= |d|τ_N. (73)

Proof. (i) First we establish an upper bound. Whend =1, following a construction similar to the one used in the proof of Lemma2.1, we define a mapU_pby

U_p(x)= _x

rp, |x|< r_p,

|xx|, |x|r_p, (74)

withr_p:= ^√N

1−^lnp^p

. A direct computation shows that forpN+1 we have

E_p(U_p)1 2

B(0,rp)

1− |U_p|²2

+Clnp p =1

2

√N

0

1−r²

N 2

N ω_Nr^N−1dr+Clnp p

= 4ωN

(N+2)(N+4)N^N/2+Clnp

p . (75)

Next we turn to the cased >1. Fixddistinct pointsq₁, . . . , q_dinR^Nwith δ:=1

4min

|q_i−q_j|: i=j

>4√ N .

FixKsatisfying K > max

1jd|q_j| +4δ,