Nondegeneracy of half-harmonic maps from $\mathbb{R}$ into $\mathbb{S}^1$

0.12 MB·

by  Y. Sire

NONDEGENERACY OF HALF-HARMONIC MAPS
FROM R INTO S1
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1 YANNICK SIRE, JUNCHENG WEI, AND YOUQUAN ZHENG
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2

n
Abstract
a
J We prove that the standard half-harmonic map U : R → S1 defined
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by
1
x2−1
P] x → x−22+x1
(cid:18)x2+1(cid:19)
A
is nondegenerate in the sense that all bounded solutions of the
.
h linearized half-harmonic map equation are linear combinations of
at threefunctionscorresponding torigidmotions(dilation, translation
m and rotation) of U.
[

1
v
1. Introduction
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2
Due to their importance in geometry and physics, the analysis of
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3 critical points of conformal invariant Lagrangians has attracted much
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attention since 1950s. A typical example is the Dirichlet energy which
.
1 is defined on two-dimensional domains and its critical points are har-
0
monic maps. This definition can be generalized to even-dimensional
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1 domains whose critical points are called polyharmonic maps. In recent
:
v years, people are very interested in the analog of Dirichlet energy in
Xi odd-dimensional case, for example, [2], [3], [4], [5], [13], [14] and the
references therein. Among these works, a special case is the so-called
r
a half-harmonic maps from R into S1 which are defined as critical points
of the line energy
1
L(u) = |(−∆R)14u|2dx. (1.1)
2
R
Z
Note that the functional L is invariant under the trace of conformal
maps keeping invariant the half-space R2: the M¨obius group. Half-
+
harmonic maps have close relations with harmonic maps with partially
free boundary and minimal surfaces with free boundary, see [12] and
[13]. Computing the associated Euler-Lagrange equation of (1.1), we
obtain that if u : R → S1 is a half-harmonic map, then u satisfies the
1
2 Y. SIRE,J. WEI, ANDY.ZHENG
following equation,
1 |u(x)−u(y)|2
(−∆R)21u(x) = dy u(x) in R. (1.2)
2π |x−y|2
R
(cid:18) Z (cid:19)
It was proved in [13] that
Proposition 1.1. ([13]) Let u ∈ H˙ 1/2(R,S1) be a non-constant entire
half-harmonic map into S1 and ue be its harmonic extension to R2.
+
Then there exist d ∈ N, ϑ ∈ R, {λ }d ⊂ (0,∞) and {a }d ⊂ R
k k=1 k k=1
such that ue(z) or its complex conjugate equals to
d
λ (z −a )−i
eiϑ k k .
λ (z −a )+i
k k
k=1
Y
Furthermore,
1
E(u,R) = [u]2 = |∇ue|2dz = πd.
H1/2(R) 2
R2
Z +
This proposition shows that the map U : R → S1
x2−1
x → x2+1
−2x
(cid:18)x2+1(cid:19)
is a half-harmonic map corresponding to the case ϑ = 0, d = 1, λ = 1
1
and a = 0. In this paper, we prove the nondegeneracy of U which is
1
a crucial ingredient when analyzing the singularity formation of half-
harmonicmapflow. NotethatU isinvariantunder translation, dilation
cosα −sinα
androtation, i.e., forQ = ∈ O(2),q ∈ Randλ ∈ R+,
sinα cosα
(cid:18) (cid:19)
the function
x−q cosα −sinα x−q
QU = U
λ sinα cosα λ
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
still satisfies (1.2). Differentiating with α, q and λ respectively and
then set α = 0, q = 0 and λ = 1, we obtain that the following three
functions
2x −4x −4x2
Z (x) = x2+1 , Z (x) = (x2+1)2 , Z (x) = (x2+1)2 (1.3)
1 x2−1 2 2(1−x2) 3 2x(1−x2)
(cid:18)x2+1(cid:19) (x2+1)2! (x2+1)2 !
satisfy the linearized equation at the solution U of (1.2) defined as
1 |U(x)−U(y)|2
1
(−∆R)2v(x) = dy v(x)
2π |x−y|2
R
(cid:18) Z (cid:19)
1 (U(x)−U(y))·(v(x)−v(y))
+ dy U(x) (i1n.4R)
π |x−y|2
R
(cid:18) Z (cid:19)
NONDEGENERACY OF HALF-HARMONIC MAPS FROM R INTO S13
for v : R → T S1. Our main result is
U
Theorem 1.1. The half-harmonic map U : R → S1
x2−1
x → x2+1
−2x
(cid:18)x2+1(cid:19)
is nondegenerate in the sense that all bounded solutions of equation
(1.4) are linear combinations of Z , Z and Z defined in (1.3).
1 2 3
In the case of harmonic maps from two-dimensional domains into
S2, the non-degeneracy of bubbles was proved in Lemma 3.1 of [7].
Integro-differential equations have attracted substantial research in re-
cent years. The nondegeneracy of ground state solutions for the frac-
tional nonlinear Schro¨dinger equations has been proved by Frank and
Lenzmann [10], Frank, Lenzmann and Silvestre [11], Fall and Valdinoci
[9], and the corresponding result in the case of fractional Yamabe prob-
lem was obtained by D´avila, del Pino and Sire in [6].
2. Proof of Theorem 1.1
The rest of this paper is devoted to the proof of Theorem 1.1. For
convenience, we identify S1 with the complex unite circle. Since Z ,
1
Z and Z are linearly independent and belong to the space L∞(R)∩
2 3
Ker(L ), weonlyneedtoprovethatthedimensionofL∞(R)∩Ker(L )
0 0
is 3. Here the operator L is defined as
0
1 |U(x)−U(y)|2
1
L0(v) = (−∆R)2v(x)− 2π |x−y|2 dy v(x)
R
(cid:18) Z (cid:19)
1 (U(x)−U(y))·(v(x)−v(y))
− dy U(x),
π |x−y|2
R
(cid:18) Z (cid:19)
4 Y. SIRE,J. WEI, ANDY.ZHENG
for v : R → T S1. Let us come back to equation (1.4), for v : R →
U
T S1, v(x)·U(x) = 0 holds pointwisely. Using this fact and the defini-
U
1
tion of (−∆R)2 (see [8]), we have
1 |U(x)−U(y)|2
1
(−∆R)2v(x) = dy v(x)
2π |x−y|2
R
(cid:18) Z (cid:19)
1 (U(x)−U(y))·(v(x)−v(y))
+ dy U(x)
π |x−y|2
R
(cid:18) Z (cid:19)
1 |U(x)−U(y)|2
= dy v(x)
2π |x−y|2
R
(cid:18) Z (cid:19)
1 (U(x)−U(y))
+ dy ·v(x) U(x)
π |x−y|2
R
(cid:18) Z (cid:19)
1 (v(x)−v(y))
+ dy ·U(x) U(x)
π |x−y|2
R
(cid:18) Z (cid:19)
1 |U(x)−U(y)|2
= dy v(x)
2π |x−y|2
R
(cid:18) Z (cid:19)
1 (v(x)−v(y))
+ dy ·U(x) U(x)
π |x−y|2
R
(cid:18) Z (cid:19)
1 |U(x)−U(y)|2
= dy v(x)
2π |x−y|2
R
(cid:18) Z (cid:19)
1
+ (−∆R)2v(x)·U(x) U(x).
(cid:16) (cid:17)
Therefore equation (1.4) becomes to
1 |U(x)−U(y)|2
1 1
(−∆R)2v(x) = dy v(x)+ (−∆R)2v(x)·U(x) U(x)
2π |x−y|2
R
(cid:18) Z (cid:19)
(cid:16) (cid:17)
2
1
= v(x)+ (−∆R)2v(x)·U(x) U(x). (2.1)
x2 +1
(cid:16) (cid:17)
Next, wewill liftequation(2.1)toS1 viathestereographic projection
from R to S1 \{pole}:
2x
S(x) = x2+1 . (2.2)
1−x2
(cid:18)x2+1(cid:19)
It is well known that the Jacobian of the stereographic projection is
2
J(x) = .
x2 +1
For a function ϕ : R → R, define ϕ˜ : S1 → R by
ϕ(x) = J(x)ϕ˜(S(x)). (2.3)
NONDEGENERACY OF HALF-HARMONIC MAPS FROM R INTO S15
Then we have
1 ϕ˜(S(x))−ϕ˜(S(y))
1
[(−∆S1)2ϕ˜](S(x)) = dS(y)
π |S(x)−S(y)|2
R
Z
1 1+x2ϕ(x)− 1+y2ϕ(y) 2
= 2 2 dy
π 4(x−y)2 1+y2
R
Z (x2+1)(y2+1)
1+x2 (1+x2)ϕ(x)−(1+y2)ϕ(y)
= dy
4π (x−y)2
R
Z
1+x2 x2 +1
= (−∆R)1/2 ϕ(x)
2 2
(cid:20) (cid:21)
1+x2
= (−∆R)1/2[ϕ˜(S(x))].
2
Therefore,
(−∆R)1/2[ϕ˜(S(x))] = J(x)[(−∆S1)12ϕ˜](S(x)).
Denote v = (v ,v ) and let v˜ , v˜ be the functions defined by (2.3)
1 2 1 2
respectively. Then the linearized equation (2.1) becomes
J(x)(−∆S1)21v˜1 = J(x)v˜1 + xx22+−11xx22+−11J(x)(−∆S1)21v˜1 + xx22−+11x−22+x1J(x)(−∆S1)21v˜2,
( J(x)(−∆S1)12v˜2 = J(x)v˜2 + x−22+x1xx22+−11J(x)(−∆S1)21v˜1 + x−22+x1x−22+x1J(x)(−∆S1)21v˜2.
Since J(x) > 0 and set U = (cosθ,sinθ), we get
(−∆S1)12v˜1 = v˜1 +cos2θ(−∆S1)21v˜1 +cosθsinθ(−∆S1)21v˜2,
(cid:26) (−∆S1)12v˜2 = v˜2 +cosθsinθ(−∆S1)21v˜1 +sin2θ(−∆S1)12v˜2,
which is equivalent to
1 1 1
(−∆S1)2v˜1 = 2v˜1 +cos2θ(−∆S1)2v˜1 +sin2θ(−∆S1)2v˜2,
1 1 1
(cid:26) (−∆S1)2v˜2 = 2v˜2 +sin2θ(−∆S1)2v˜1 −cos2θ(−∆S1)2v˜2.
Set w = v˜ +iv˜ , z = cosθ+isinθ, then we have
1 2
(−∆S1)21w = 2w+z2(−∆S1)21w¯. (2.4)
Here w¯ is the conjugate of w.
Since v ∈ L∞(R), w is also bounded, so we can expand w into fourier
series
∞
w = a zk.
k
k=−∞
X
6 Y. SIRE,J. WEI, ANDY.ZHENG
1
Note that all the eigenvalues for (−∆S1)2 are λk = k, k = 0,1,2,·· ·,
see [1]. Using (2.4), (−∆S1)12zk = kzk and (−∆S1)21z¯k = kz¯k, we obtain
(−k −2)ak = (2−k)a¯2−k, if k < 0,
(k −2)ak = (2−k)a¯2−k, if 0 ≤ k ≤ 2,

 ak = a¯2−k, if k ≥ 3.
Furthermore, fromthe orthogonal condition v(x)·U(x) = 0 (so (v˜ ,v˜ )·
 1 2
(cosθ,sinθ) = 0), we have
ak = −a¯2−k, k = ···−1,0,1,··· .
Thus
a = 0, if k < 0 or k ≥ 3
k
and
a = −a¯ , a = −a¯
0 2 1 1
hold, which imply that
i (z2 −1)
w = −a¯ +a z +a z2 = a(iz)+b (z −1)2 +c .
2 1 2
2 2
(cid:20) (cid:21)
Here a, b, c are real numbers and satisfy relations
c i
i(a−b) = a , + b = a .
1 2
2 2
And it is easy to check that iz, i(z − 1)2 and (z2−1) are respectively
2 2
Z , Z and Z under stereographic projection (2.2). By the one-to-one
1 2 3
correspondence of w and v, we know that the dimension of L∞(R) ∩
Ker(L ) is 3. This completes the proof.
0
References
[1] Sun-Yung A. Chang and Paul C. Yang. Extremal metrics of zeta function
determinants on 4-manifolds. Ann. of Math. (2), 142(1):171–212,1995.
[2] Francesca Da Lio. Fractional harmonic maps into manifolds in odd dimension
n>1. Calc. Var. Partial Differential Equations, 48(3-4):421–445,2013.
[3] Francesca Da Lio. Compactness and bubble analysis for 1/2-harmonic maps.
Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 32(1):201–224,2015.
[4] Francesca Da Lio andTristanRivi`ere.Sub-criticality of non-localSchr¨odinger
systemswithantisymmetricpotentialsandapplicationstohalf-harmonicmaps.
Adv. Math., 227(3):1300–1348,2011.
[5] Francesca Da Lio and Tristan Rivi`ere.Three-term commutator estimates and
1
the regularity of -harmonic maps into spheres. Anal. PDE, 4(1):149–190,
2
2011.
[6] Juan D´avila, Manuel del Pino and Yannick Sire. Nondegeneracy of the bub-
ble in the critical case for nonlocal equations. Proc. Amer. Math. Soc.,
141(11):3865–3870,2013.
NONDEGENERACY OF HALF-HARMONIC MAPS FROM R INTO S17
[7] Juan D´avila, Manuel del Pino and Juncheng Wei. Singularity formation for
2
the two-dimensional harmonic map flow into S . Preprint.
[8] Eleonora Di Nezza, Giampiero Palatucci and Enrico Valdinoci. Hitchhiker’s
guide to the fractionalSobolevspaces.Bull. Sci. Math., 136(5):521–573,2012.
[9] Mouhamed Moustapha Fall and Enrico Valdinoci. Uniqueness and nondegen-
eracy of positive solutions of (−∆)su+u = up in RN when s is close to 1.
Comm. Math. Phys., 329(1):383–404,2014.
[10] Rupert L.FrankandEnno Lenzmann.Uniqueness ofnon-lineargroundstates
for fractional Laplacians in R. Acta Math., 210(2):261–318,2013.
[11] Rupert L. Frank, Enno Lenzmann and Luis Silvestre. Uniqueness of radial
solutions for the fractional Laplacian. Comm. Pure Appl. Math., 69(9):1671–
1726, 2016.
[12] Ailana FraserandRichardSchoen.Sharpeigenvaluebounds andminimalsur-
faces in the ball. Invent. Math., 203(3):823–890,2016.
[13] Vincent Millot and Yannick Sire. On a fractional Ginzburg-Landau equation
and 1/2-harmonic maps into spheres. Arch. Ration. Mech. Anal., 215(1):125–
210, 2015.
[14] ArminSchikorra.Regularityofn/2-harmonicmapsintospheres.J.Differential
Equations, 252(2):1862–1911,2012.
JohnsHopkinsUniversity,Departmentofmathematics,KriegerHall,
Baltimore, MD 21218, USA
E-mail address: [email protected]
Department of Mathematics, University of British Columbia, Van-
couver, B.C., Canada, V6T 1Z2
E-mail address: [email protected]
SchoolofScience,TianjinUniversity,92WeijinRoad,Tianjin300072,
P.R. China
E-mail address: [email protected]