Compactness and bubble analysis for 1/2-harmonic maps

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Ann. I. H. Poincaré – AN 32 (2015) 201–224

www.elsevier.com/locate/anihpc

Compactness and bubble analysis for 1/2-harmonic maps

Francesca Da Lio

Department of Mathematics, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland

Received 10 October 2012; received in revised form 30 September 2013; accepted 12 November 2013 Available online 1 December 2013

Abstract

In this paper we study compactness and quantization properties of sequences of 1/2-harmonic mapsu_k:R→S^m−1such that uk_H˙1/2(R,S^m−1)C. More precisely we show that there exist a weak 1/2-harmonic mapu_∞:R→S^m−1, a finite and possible empty set{a1, . . . , a} ⊂Rsuch that up to subsequences

(−)^1/4u_k²dx (−)^1/4u_∞²dx+

i=1

λ_iδ_ai, in Radon measure,

ask→ +∞, withλ_i0.

The convergence ofu_ktou_∞is strong inW˙_loc^1/2,p(R\ {a₁, . . . , a}), for everyp1. We quantify the loss of energy in the weak convergence and we show that in the case of non-constant 1/2-harmonic maps with values inS¹one hasλ_i=2π n_i, withn_i a positive integer.

©2013 Elsevier Masson SAS. All rights reserved.

MSC:58E20; 35J20; 35B65; 35J60; 35S99

Keywords:Fractional harmonic maps; Nonlinear elliptic PDE’s; Regularity of solutions; Commutator estimates

1. Introduction

In the paper[9]Rivière and the author started the investigation of the following 1-dimensional quadratic Lagrangian L(u)=

R

(−)^1/4u(x)²dx, (1) whereu:R→N,N is a smooth k-dimensional sub-manifold ofR^m which is at leastC², compact and without boundary. We observe that(1)is a simple model of Lagrangian which is invariant under the trace of conformal maps that keep invariant the half-spaceR²₊: the Möbius group.

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

http://dx.doi.org/10.1016/j.anihpc.2013.11.003

Precisely letφ:R²₊→R²₊inW^1,2(R²₊,R²₊)be a conformal map of degree 1, i.e. it satisfies

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩ ∂φ

∂x =

∂φ

∂y ,

∂φ

∂x,∂φ

∂y

=0,

det∇φ0 and ∇φ=0.

(2)

Here·,· denotes the standard Euclidean inner product inR^m.

We denote byφ˜the restriction ofφtoR. Then we haveL(u◦ ˜φ)=L(u).

MoreoverL(u)in(1)coincides with the semi-normu²_H˙1/2(R)and the following identity holds

R

(−)^1/4u(x)²dx=inf

R²₊

|∇ ˜u|²dx: u˜∈W^1,2

R²,R^m

, traceu˜=u

. (3)

The LagrangianLextends to mapsuin the following function space H˙^1/2(R,N)=

u∈ ˙H^1/2 R,R^m

: u(x)∈N,a.e.

.

The operator(−)^1/4onRis defined by means of the Fourier transform as follows (−)^1/4u= |ξ|^1/2uˆ

(given a functionf,fˆdenotes the Fourier transform off).

We denote byπ_N the orthogonal projection fromR^montoN which happens to be aCmap in a sufficiently small neighborhood ofN ifN is assumed to beC⁺¹. We now introduce the notion of 1/2-harmonic map into a manifold.

Definition 1.1. A mapu∈ ˙H^1/2(R,N)is called a weak 1/2-harmonic map intoN if for anyφ∈ ˙H^1/2(R,R^m)∩ L^∞(R,R^m)there holds

d dtL

π_N(u+t φ)

|t=0=0.

In short we say that a weak 1/2-harmonic map is acritical point ofL inH˙^1/2(R,N)for perturbations in the target.

We next give some geometric motivations related to the study of the problem(1).

First of all variational problems of the form(1)appear as a simplified model of renormalization area in hyperbolic spaces, see for instance[2]. There are also some geometric connections which are being investigated in the paper[11]

between 1/2-harmonic maps and the so-called free boundary sub-manifolds and optimization problems of eigenval- ues. With this regards we refer the reader also to the papers[14,15]. Finally 1/2-harmonic maps into the circleS¹ might appear for instance in the asymptotics of equations in phase-field theory for fractional reaction–diffusion such as

²(−)^1/2u+u

1− |u|²

=0,

whereuis a complex-valued “wave function”.

In this paper we consider the caseN=S^m−1. It can be shown (see[9]) that every weak 1/2-harmonic map satisfies the following Euler–Lagrange equation

(−)^1/2u∧u=0 inD(R). (4)

One of the main achievements of the paper[9]is the rewriting of Eq.(4)in a more “tractable” way in order to be able to investigate regularity and compactness property of weak 1/2-harmonic maps. Precisely in[9]the following two results have been proved:

Proposition 1.1. A mapu inH˙^1/2(R,S^m−1)is a weak1/2-harmonic map if and only if it satisfies the following equation

(−)^1/4

u∧(−)^1/4u

=T

ω(u), u

, (5)

whereω(u):R^m→2R^m,v→ω(u)v:=u∧v and in general for arbitrary positive integersn,m,, for every linear operatorQ∈ ˙H^1/2(Rⁿ,M×m(R))¹andu∈ ˙H^1/2(Rⁿ,R^m),T is the operator defined by

T (Q, u):=(−)^1/4

Q(−)^1/4u

−Q(−)^1/2u+(−)^1/4Q(−)^1/4u. (6) Eq.(5)has been completed by the following “structure equation” which is a consequence of the fact thatu∈S^m−1 almost everywhere:

Proposition 1.2.All maps inH˙^1/2(R,S^m−1)satisfy the following identity (−)^1/4

u·(−)^1/4u

=S

ω(u), u

−R

(−)^1/4u·R(−)^1/4u

, (7)

whereω(u):R^m→R,v→ω(u)v:=u·v and in general for arbitrary positive integersn,m,, for everyQ∈ H˙^1/2(Rⁿ,M×m(R)), andu∈ ˙H^1/2(Rⁿ,R^m),Sis the operator given by

S(Q, u):=(−)^1/4

Q(−)^1/4u

−R(Q∇u)+R

(−)^1/4QR(−)^1/4u

(8) andRis the Fourier multiplier of symbolm(ξ )= −i_|^ξ_ξ|.

We call the operatorsT,S three-term commutatorsand in[9]the following estimates have been established: for everyu∈ ˙H^1/2(R,R^m)andQ∈ ˙H^1/2(R,M×m(R))we have

T (Q, u)_˙

H^−1/2(R)CQ_H˙^1/2_(R)u_H˙^1/2_(R), (9)

S(Q, u)_˙

H^−1/2(R)CQH˙^1/2(R)uH˙^1/2(R), (10)

and R

(−)^1/4u·R(−)^1/4u_˙

H^−1/2(R)Cu²_H˙1/2(R). (11)

What has been discovered is a sort of “gain of regularity” in the r.h.s. of Eqs.(5)and(7)in the sense that, under the assumptionsu∈ ˙H^1/2(R,R^m)andQ∈ ˙H^1/2(R,M×m(R))each term individually inT andS– like for instance (−)^1/4[Q(−)^1/4u]orQ(−)^1/2u . . .– is not inH˙^−1/2but the special linear combination of them constitutingT andSis inH˙^−1/2. The same phenomenon appears in dimension 2 in the context of harmonic maps, for the Jacobians J (a, b):=^∂a_∂x^∂b_∂y −^∂a_∂y^∂b_∂x (witha, b∈ ˙H¹(R²)) which satisfy as a direct consequence of Wente’s theorem (see[5,25])

J (a, b)_˙

H⁻¹(R²)CaH˙¹(R²)bH˙¹(R²) (12)

whereas, individually, the terms ^∂a_∂x^∂b_∂y and ^∂a_∂y^∂b_∂x are not inH˙⁻¹(R²).

The estimates(9)and(10)imply in particular that ifu∈ ˙H^1/2(R,S^m−1)is a 1/2-harmonic map then (−)^1/4u

L²(R)C(−)^1/4u²

L²(R), (13)

where the constantCis independent ofu.

From the inequality (13) it follows that if C(−)^1/4uL²(R)<1 then the solution is constant. This the so- calledbootstrap testand it is the key observation to prove Morrey-type estimates and to deduce Hölder regularity of 1/2-harmonic maps, see[9].

We mention here that since the paper[9]several extensions have been considered. The regularity of solutions to nonlocal linear Schrödinger systems with applications to 1/2-harmonic maps with values into general manifolds have

1 M×m(R)denotes, as usual, the space of×mreal matrices.

been studied by Rivière and the author in[10].n/2-harmonic maps in odd dimensionnhas been considered in[23]

and[6] respectively in the case of values into the(m−1)-dimensional sphere and into general manifolds and the case ofα-harmonic maps inW˙^α,p(Rⁿ,S^m−1), withαp=n, has been recently studied by Schikorra and the author in[12]. Finally Schikorra[24]has also studied the partial regularity of weak solutions to nonlocal linear systems with an antisymmetric potential in the supercritical case (namely whereαp < n) under a crucial monotonicity assumption on the solutions which allows to reduce to the critical case.

In this paper we address to the issue of understanding the behavior of sequencesu_k of weak 1/2-harmonic maps.

We observe that as in the case of harmonic maps the bootstrap test(13)implies that if the energy issmallthen the system behaves locally like a linear system of the form(−)^1/2u=0 (namely the r.h.s. is “dominated” by the l.h.s. of the equation). As a consequence we obtain that any sequenceu_kof weak 1/2-harmonic maps with uniformly bounded energy weakly converges to a weak 1/2-harmonic mapu_∞and strongly converges tou_∞away from a finite (possibly empty) set{a₁, . . . , a} ⊂R.

Namely we have (up to a subsequence) (−)^1/4uk²dx (−)^1/4u_∞²dx+

i=1

λiδa_i, in Radon measure,

ask→ +∞, withλ_i0. It remains the question to understand how the convergence at the concentration pointsa_i fails to be strong. A careful analysis shows that the loss of energy during the weak convergence is not only concen- trated at the pointsa_i but it is also quantized: this amount of energy is given by the sum of energies of non-constant 1/2-harmonic maps (the so-calledbubbles). More precisely we get the following result:

Theorem 1.1.Letu_k∈ ˙H^1/2(R,S^m−1)be a sequence of1/2-harmonic maps such thatu_kH˙^1/2C. Then it holds:

1. There existu_∞∈ ˙H^1/2(R,S^m−1)and a possibly empty set{a1, . . . , a},1, such that up to subsequences un→u_∞ inW˙_loc^1/2,p

R\ {a1, . . . , a}

, p2ask→ +∞ (14)

and

(−)^1/2u_∞∧u_∞=0, inD(R). (15)

2. There is a familyu˜^i,j_∞ ∈ ˙H^1/2(R,S^m−1)of1/2-harmonic maps(i∈ {1, . . . , },j∈ {1, . . . , Ni}), such that up to subsequences

(−)^1/4

uk−u_∞−

i,j

˜ u^i,j_∞

L²_loc(R)

→0, ask→ +∞. (16)

Theorem 1.1 says that for everyi,λ_i =Ni

j=1L(u˜^i,j_∞). Therefore there is no dissipation of energy in the region betweenu_∞and the bubbles and between the bubbles themselves (the so-calledneck-regions).

We would like now to mention a result obtained in the paper[11]on the characterization of 1/2 harmonic maps u:S¹→S¹which permits us to deduce that in the case of 1/2 harmonic maps with values inS¹one hasλi=2π ni, withni a positive integer and also to provide a simple example showing that the quantization may actually occur, namely the set{a1, . . . , a}may be nonempty.

Theorem 1.2.(See[11].)u:S¹→S¹is a weak1/2-harmonic map if and only if its harmonic extensionu˜:D²→R² is holomorphic or anti-holomorphic.

We remark that because of the invariance of the Lagrangian(1)with respect to the trace of conformal transforma- tions we can study without restrictions the problem inS¹instead ofR.

From Theorem 1.2it follows that 1/2-harmonic mapsu:S¹→S¹ withdeg(u)=1 coincide with the trace of Möbius transformations of the diskD²⊆R². Moreover every non-constant weak 1/2-harmonic mapu:S¹→S¹ satisfies

S¹

(−)^1/4u²dx=2π k <+∞,

wherekis a positive integer which coincides with|deg(u)|.

Let us consider now the following sequence of 1/2-harmonic maps u_n:S¹→S¹, u_n(z)= z−an

1− ¯anz,

with|a_n|<1 anda_n→1 asn→ +∞. In this case we haveu_n→ −1 inC_loc^∞(R\ {1}), thus the set of concentration points is nonempty.Theorem 1.1yields the existence of one bubbleu˜_∞such that

(−)^1/4(u_n− ˜u_∞)

L²_loc→0, asn→ +∞.

We explain now the method we have used to prove the mainTheorem 1.1.

In order to get the quantization of the energy we exploit a “functional analysis” method introduced by Lin and Rivière in [20] in the context of harmonic maps in non-conformal dimensions, i.e., in dimension n3. Such a method consists in the use of the interpolation Lorentz spaces in the special case where the r.h.s. of the equation can be written as a linear combination of Jacobians.

This technique has been recently applied in[18,19]and in[3]for the quantization analysis respectively of linear Schrödinger systems with antisymmetric potential in 2-dimension, of bi-harmonic maps in 4-dimensions, and of Willmore surfaces. We refer the reader to the papers[21,20]for an overview of the bubbling and quantization issues in the literature.

We describe briefly the key steps to get the quantization analysis.

1. First of all we will make use of a general result proved in[3]which permits to split the domain (in our caseR) into the converging region(which is the complement of small neighborhoods of theaⁱ),bubbles domainsandneck-regions (which are unions of degenerating annuli).

2. We prove that theL² norm of (−)^1/4u_k in the neck regions is arbitrary small (seeTheorem 3.1). Thanks to the duality of the Lorentz spacesL^2,1−L^2,∞this is reduced in estimating theL^2,∞, L^2,1norms ofu_k in these regions. Precisely we first show that the L^2,∞ norm of theu_k is arbitrary small indegenerating annuli, i.e. annuli whose conformal class is not a priori bounded (seeLemma 3.2) and as far as theL^2,1norm is concerned, we use the following improved estimate on the operatorsT andSwhich is proved inAppendix A.

Theorem 1.3.Letu, Q∈ ˙H^1/2(Rⁿ). ThenT (Q, u), S(Q, u)∈H¹(Rⁿ)and T (Q, u)

H¹(Rⁿ)CQH˙^1/2(Rⁿ)uH˙^1/2(Rⁿ), (17) S(Q, u)

H¹(Rⁿ)CQH˙^1/2(Rⁿ)uH˙^1/2(Rⁿ). (18) We recall here some definitions. We denote byL^2,∞(Rⁿ)the space of measurable functionsf such that

sup

λ>0

λx∈Rⁿ: f (x)λ^1/2<+∞,

andL^2,1(Rⁿ)is the space of measurable functions satisfying

+∞

0

x∈R: f (x)λ^1/2dλ <+∞.

We denote byH¹(Rⁿ)the Hardy space which is the space ofL¹functionsf onRⁿsatisfying

Rⁿ

sup

t >0

|φt∗f|(x) dx <+∞,

whereφ_t(x):=t⁻ⁿφ(t⁻¹x)andφis some function in the Schwartz spaceS(Rⁿ)satisfying

Rⁿφ(x) dx=1.

Finally for everys∈Randq >1 we denote byW˙^s,q(Rⁿ)the fractional Sobolev space f ∈S

Rⁿ : F⁻¹

|ξ|^sF[f]

∈L^q Rⁿ

.

For more properties on the Lorentz spaces, Hardy spaceH¹and fractional Sobolev spaces we refer to[16]and[17].

In a forthcoming paper [8] we are going to investigate bubbles and quantization issues in the case of nonlocal Schrödinger linear systems with applications to 1/2-harmonic maps with values into manifolds. The difficulty there is to succeed in getting a uniformL^2,1estimate as well on degenerating annuli as in the local case (see[18]).

It would be also very interesting to understand the geometric properties of the bubbles in the case of more general manifolds.

This paper is organized as follows.

Section2we address to the compactness issue which is the first part ofTheorem 1.1. In Section3 we proveL² estimates on degenerating annual domains. In Section4we prove the second part ofTheorem 1.1. InAppendix Awe proveTheorem 1.3.

2. Compactness

In this section we prove the first part ofTheorem 1.1. The result is based on the followingε-regularityproperty whose proof can be found in[9,10]and in[7].

Lemma 2.1(ε-regularity). Letu∈ ˙H^1/2(R,S^m−1)be a1/2-harmonic map. Then there existsε₀>0such that if

j0

2^−j/2(−)^1/4u

L²(B(x,2^jr))ε0, (19)

then there isp >2 (independent ofu)such that for everyx∈R,y∈B(x, r/2)we have

r^p/2−1

B(y,r/2)

(−)^1/4u(x)^pdx 1/p

C

j0

2^−j/2(−)^1/4u

L²(B(x,2^jr)), (20)

whereC >0depends onuH˙^1/2u(R).

By bootstrapping into Eqs.(5)and(7)and by localizingTheorems A.1 and A.2(see[7]) one can show the follow- ing:

Corollary 2.1.Letu∈ ˙H^1/2(R,S^m−1)be a1/2-harmonic map. Then for everyx∈Rthere existsr >0such that the following holds

r^1/2−1/p(−)^1/4u

L^p(B(x,r))CuH˙^1/2(R), (21)

for everyp2and r^1/2(−)^1/4u

L∞(B(x,r))Cu_H˙^1/2_(R). (22)

We will use also the localized version of the following result whose proof can be found in[1, p. 78]:

Lemma 2.2.Let0< α <1andg∈L^p(R),1< p <∞. Then there is a constantC >0independent ofg, such that (−)^−αθ2 g

L^r(Rⁿ)C(−)^−α2g^θ

L^s(Rⁿ)g¹_L⁻p^θ(Rⁿ), for0< θ <1,1s∞,¹_r =^θ_s +¹⁻_p^θ.

Next we show that if for somex∈Randρ >0 we have

j0

2^−j/2(−)^1/4u

L²(B(x,2^jρ))ε₀,

whereε₀>0 is the small constant appearing in theε-regularityLemma 2.1, thenu∈ ˙W

1 q+¹2,2

loc (B(x, ρ)), for allq2, i.e.(−)^2q1+14u∈L²_loc(B(x, ρ)).

Proposition 2.1.Letx∈Randρ >0be such that

j0

2^−j/2(−)^1/4u

L²(B(x,2^jρ))ε₀

withε₀>0given inLemma2.1. Then there existsC >0 (independent ofx,ρand dependent onu_H˙1/2(R))such that for allq >2the following holds

(−)^2q1+14u

L²(B(y,ρ/2))C

j0

2^−j/2(−)^1/4u

L²(B(x,2^jρ)), (23)

for ally∈B(x, ρ/2).

Proof. We setv=(−)^1/4u. FromLemma 2.1it follows that there exists q >2 (independent ofu) such that for everyy∈B(x, ρ/2)

vL^q(B(y,ρ/4))C

j0

2^−j/2(−)^1/4u

L²(B(x,2^jρ)). (24)

By bootstrapping into Eqs.(5)and(7)one gets (−)^1/4v

L

2q

q+2(B(y,ρ/4))C

j0

2^−j/2(−)^1/4u

L²(B(x,2^jρ)). (25)

Now we setf :=(−)^1/4v. By applyingLemma 2.2inB(y, ρ/4)withg:=v,p=q,r=2,s=_q^2q₊₂,α=¹₂ and θ=^q−_q²we obtain

(−)^−q−24q f

L²(B(y,ρ/4))Cf^θ

L

q+22q (B(y,ρ/4))

v¹_L^−qθ_(B(y,ρ/4)). (26)

In particular we get that(−)^2q1v∈L²(B(y, ρ/4))and hence(−)^2q1+14u∈L²(B(y, ρ/4))with (−)^2q1+14u

L²(B(y,ρ/4))C

j0

2^−j/2(−)^1/4u

L²(B(x,2^jρ)). (27)

FromCorollary 2.1it follows that the above arguments actually hold for everyq2. This concludes the proof. 2 We show now a singular point removability type result for 1/2-harmonic maps.

Proposition 2.2(Singular point removability). Letu∈ ˙H¹²(R,S^m−1)be a1/2-harmonic map inD(R\ {a1, . . . , a}).

Then

u∧(−)¹²u=0 inD(R).

Proof. The fact that

u∧(−)¹²u=0 inD

R\ {a₁, . . . , a} implies that

(−)^1/4

u∧(−)^1/4u

=T (u∧, u) inD

R\ {a₁, . . . , a} , whereT (u∧, u)∈ ˙H⁻¹²(R)and

T (u∧, u)

˙

H⁻¹2(R)Cu²

˙ H¹2(R)

. (28)

The distributionφ:=(−)^1/4(u∧(−)^1/4u)−(T (u∧, u))is of orderp=1 and supported in{a₁, . . . , a}. Therefore by Schwartz Theorem[4]one has

φ=

|α|1

c_α∂^αδ_ai.

Sinceφ∈ ˙H⁻¹²(R), then the above implies thatc_α=0 and thus (−)^1/4

u∧(−)^1/4u

=T (u∧, u) inD(R).

We conclude the proof ofProposition 2.2. 2

The proof of thefirst part ofTheorem 1.1concerning the compactness of uniformly bounded 1/2-harmonic maps is contained in the following lemma.

Lemma 2.3.Letu_k∈ ˙H^1/2(R,S^m−1)be a sequence of1/2-harmonic maps such thatu_kH˙^1/2C. Then there exist a subsequenceu_k ofu_k, a functionu_∞∈ ˙H^1/2(R,S^m−1)and{a₁, . . . , a},1, such that

u_k→u_∞ ask→ +∞inH˙

1 2

loc

R\ {a₁, . . . , a}

forp2, (29)

and

(−)^1/2u_∞∧u_∞=0 inD(R). (30)

Proof. 1.First of all there exist a subsequenceu_k of uk, a function u_∞∈ ˙H^1/2(R,S^m−1)such thatu_k u_∞as k→ +∞.

2.Suppose first that for someρ >0 and for allk1,

j02^−j/2(−)^1/4ukL²(B(x,2^jρ))ε0withε0>0 given inLemma 2.1. Then fromLemma 2.1and the Rellich–Kondrachov Theorem (the embeddingW˙ ^1q+12,2(B(x, ρ/4)) → W˙ ^12,t(B(x, ρ/4))is compact for allt <_q^2q₋₂) it follows that

u_k→u_∞ ask→ +∞inH˙^1/2

B(x, ρ/4),S^m−1 for allx∈R. In particular we have also

(−)¹²u_k→(−)¹²u_∞ ask→ +∞inH˙^−1/2

B(x, ρ/4),S^m−1 . Hence

(−)¹²u_k∧u_k→(−)¹²u_∞∧u_∞ ask→ +∞inD

B(x, ρ/4) , and

(−)¹²u_∞∧u_∞=0 inD

B(x, ρ/4) .

3. Claim 1.There are only finitely many points{a₁, . . . , a}such that (−)¹²u_∞∧u_∞=0 inD

R\ {a1, . . . , a}

. (31)

Proof of Claim 1. We associate to everyxthe numberρ_x^k>0 such that (−)^1/4u_k

L²(B(x,ρ_x^k))=ε₀ 8 whereε₀is as inLemma 2.1.

For everyM >0 andk1 we set I_k^M:=

x: ρ^k_x< 1 M

and

Fk^M:=

B

x, 1

M

, x∈I_k^M

.

By the Vitali–Besicovitch Covering Theorem (see for instance [13]), we can find an at most countable family of points(x_j^k,M)_j∈JM

k ,x_j^k,M∈I_k^M andI_k^M ⊆

j∈J_k^MB(x_j^k,M,_M¹). Moreover everyx∈F_k^M is contained in at mostN balls,Nbeing a number depending only on the dimension of the space.

Now we observe that

Cuk²_H˙1/2(R)=(−)^1/4uk²

L²(R)

N⁻¹

j∈J_k^M

(−)^1/4u_k²

L²(B(x^k,M_j ,_M¹))

N⁻¹

j∈J_k^M

ε²₀

64=N⁻¹J_k^Mε²₀ 64.

Thus|J_k^M|<+∞for everykandMand this implies that forkandMlarge enough|J_k^M| =C, withCindependent ofkandM. In particular there existsm₀>0 such that

I_k^M⊆

m0

j=1

B

x_j^k,M, 1 M

.

By definition we haveI_k^M+1⊆I_k^M for allk andM. By using adiagonal procedurewe can subtract a subsequence k→ +∞such thatx_j^k,M→x_j^∞,Mfor allM >0 andj and

I_∞^M⊆

m₀

j=1

B

x_j^∞,M, 1 M

.

Now we letM→ +∞and get I_∞⁰ ⊆J_∞,0:=

x_j^∞,0

j=1,...m₀. 2

Claim 2.Ifx /∈J_∞,0then there existsr >˜ 0such that u_∞∧(−)¹²u_∞=0 inD

B(x,r)˜ .

Proof of Claim 2. We assume thatx_j^∞,0= ∞for allj =1, . . . , m0. Letγ=dist(x, J_∞,0)andK >0 be such that 2K⁻¹< γ. LetM >˜ 0 be such that for allMM˜ and for allj =1, . . . , n0we have

x_j^∞,0−x_j^∞,M< 1

4K. (32)

Letk >¯ 0 be such that for allkk¯and for allj=1, . . . , m0we have x^∞,M

j −x_j^k,M< 1

4K. (33)

By combining(32)and(33)we get x−x^k_j^,Mx−x_j^∞,0

−x_j^∞,0−x_j^∞,M−x_j^∞,M˜ −x_j^k,M

2

K− 1 4K − 1

4K = 3 2K > 1

M. Therefore x /∈ m₀

j=1B(x^k_j^,M,_M¹), and x /∈ I_k^M for all k k. In particular¯ ρ_k,x M⁻¹ and (up to subse- quences) ρ_k,x → ρ_∞,x >0. Now let 0 < r < ρ_∞,x. We observe that B(x, r) ⊆B(x, ρ_k,x), for k large and (−)^1/4u_kL²(B(x,r))^ε₈⁰.

Letj₀3 be such that 2^−j0/2(

R|(−)^1/4u_k|²dx)^1/2ε₈. We now estimate

h02^−h/2(−)^1/4u_kL2(B(x,2^h(2^−2j0r))).

h0

2^−h/2(−)^1/4u_k

L²(B(x,2^h(2^−2j0r)))

=

j₀

h=0

2^−h/2(−)^1/4u_k

L²(B(x,2^h(2^−2j0r)))+ ∞ h=j0+1

2^−h/2(−)^1/4u_k

L²(B(x,2^h(2^−2j0r)))

ε0

8 _∞

h=0

2^−h/2

+2^−(j0+1)/2

R

(−)^1/4u_k²dx 1/2

√2

√2−1 ε 8 + ε

16< ε0. By applying Step 2 we get

u_∞∧(−)^1/2u_∞=0 inD B(x,r)˜

forr <˜ 2^−2j0r. This concludes the proof ofClaim 2by settinga_i:=x_i^∞and=m₀. 2 Now we applyProposition 2.2and we get that

u_∞∧(−)^1/2u_∞=0 inD(R).

We can conclude the proof ofLemma 2.3and the first part ofTheorem 1.1. 2 3. L^2,∞andL²estimates on degenerating annuli

The goal of this section is to prove some energy estimates of 1/2-harmonic maps in degenerating annuli, i.e.

annuli whose conformal class is not a priori bounded. Such estimates are crucial in the next section in order to get the quantization analysis in the neck regions.

The main result of this section is

Theorem 3.1.There existsδ >˜ 0such that for any1/2-harmonic mapu∈ ˙H¹²(R,S^m−1), for anyδ <δ˜andλ, Λ >0 withλ < (2Λ)⁻¹satisfying

sup

ρ∈[λ,(2Λ)−1]

B(0,2ρ)\B(0,ρ)

(−)^1/4u²dx 1/2

δ, (34)

we have

B(0,Λ⁻¹)\B(0,λ)

(−)^1/4u²dxC sup

ρ∈[λ,(2Λ)⁻¹]

B(0,2ρ)\B(0,ρ)

(−)^1/4u²dx 1/2

. (35)

The proof ofTheorem 3.1consists in three steps:

(1) First we show that we can control in degenerating annuli theL^qnorm of(−)^1/4ufor someq >2 by

sup

ρ∈[λ,(2Λ)⁻¹]

B(0,2ρ)\B(0,ρ)

(−)^1/4u²dx 1/2

. (36)

(2) Then we estimate theL^2,∞norm of(−)^1/4uin degenerating annuli in terms of(36).

(3) Finally we use the globalL^2,1estimates obtained inAppendix A(seeTheorem 1.3) and the dualityL^2,1−L^2,∞ in order to conclude.

Lemma 3.1(L^q-estimates). There existsδ >˜ 0such that for every1/2-harmonic mapu∈ ˙H¹²(R), if

sup

ρ∈[λ,(2Λ)⁻¹]

B(0,2ρ)\B(0,ρ)

(−)^1/4u²dx 1/2

δ, (37)

for someδ <δ,˜ λ, Λ >0with2λ < (4Λ)⁻¹, then there existq >2andθ∈(0,1)(independent ofu,λ,Λ)such that

sup

ρ∈[2λ,(4Λ)⁻¹]

ρ^q/2−1

B(0,2ρ)\B(0,ρ)

(−)^1/4u^qdx 1/q

C

sup

ρ∈[λ,(2Λ)−1]

B(0,2ρ)\B(0,ρ)

(−)^1/4u²dx 1/2θ

.

Proof. We chooseδ=^ε₈⁰ whereε₀>0 is the constant appearing in theε-regularityLemma 2.1.

Step 1.There existsp >2 (independent ofλ,Λ,u) such that

sup

ρ∈[2λ,(4Λ)−1]

ρ^p/2−1

B(0,2ρ)\B(0,ρ)

(−)^1/4u^pdx 1/p

Cu_˙

H¹²(R). (38)

Proof of Step 1. Letr >0 be such that

B(0,2r)\B(0,r)

(−)^1/4u²dx 1/2

< δ.

Claim.There existsp >2 (independent ofu,δandr)such that

r^p/2−1

B(0,³₂r)\B(0,⁵₄r)

(−)^1/4u^pdx 1/p

Cu_˙

H¹2(R). (39)

Lety∈B(0,³₂r)\B(0,⁵₄r)(we clearly havedist(y, ∂(B(0,2r)\B(0, r)))1/4).

Let j₀3 such that 2^−j0/2(

R|(−)^1/4u|²dx)^1/2δ, and B(y,2^−j0r)⊂(B(0,2r)\B(0, r)) for all y ∈ B(0,³₂r)\B(0,⁵₄r).