Nondegeneracy of half-harmonic maps from $\mathbb {R}$ into $\mathbb {S}^1$
HTML articles powered by AMS MathViewer
- by Yannick Sire, Juncheng Wei and Youquan Zheng PDF
- Proc. Amer. Math. Soc. 146 (2018), 5263-5268 Request permission
Abstract:
We prove that the standard half-harmonic map $U:\mathbb {R}\to \mathbb {S}^1$ defined by \begin{equation*} x\to \begin {pmatrix} \frac {x^2-1}{x^2+1} \\[4pt] \frac {-2x}{x^2+1} \end{pmatrix} \end{equation*} is nondegenerate in the sense that all bounded solutions of the linearized half-harmonic map equation are linear combinations of three functions corresponding to rigid motions (dilation, translation, and rotation) of $U$.References
- Sun-Yung A. Chang and Paul C. Yang, Extremal metrics of zeta function determinants on $4$-manifolds, Ann. of Math. (2) 142 (1995), no. 1, 171–212. MR 1338677, DOI 10.2307/2118613
- Francesca Da Lio, Fractional harmonic maps into manifolds in odd dimension $n>1$, Calc. Var. Partial Differential Equations 48 (2013), no. 3-4, 421–445. MR 3116017, DOI 10.1007/s00526-012-0556-6
- Francesca Da Lio, Compactness and bubble analysis for 1/2-harmonic maps, Ann. Inst. H. Poincaré C Anal. Non Linéaire 32 (2015), no. 1, 201–224. MR 3303947, DOI 10.1016/j.anihpc.2013.11.003
- Francesca Da Lio and Tristan Rivière, Sub-criticality of non-local Schrödinger systems with antisymmetric potentials and applications to half-harmonic maps, Adv. Math. 227 (2011), no. 3, 1300–1348. MR 2799607, DOI 10.1016/j.aim.2011.03.011
- Francesca Da Lio and Tristan Rivière, Three-term commutator estimates and the regularity of $\frac 12$-harmonic maps into spheres, Anal. PDE 4 (2011), no. 1, 149–190. MR 2783309, DOI 10.2140/apde.2011.4.149
- Juan Dávila, Manuel del Pino, and Yannick Sire, Nondegeneracy of the bubble in the critical case for nonlocal equations, Proc. Amer. Math. Soc. 141 (2013), no. 11, 3865–3870. MR 3091775, DOI 10.1090/S0002-9939-2013-12177-5
- Juan Dávila, Manuel del Pino, and Juncheng Wei, Singularity formation for the two-dimensional harmonic map flow into $S^2$, arXiv:1702.05801.
- Eleonora Di Nezza, Giampiero Palatucci, and Enrico Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), no. 5, 521–573. MR 2944369, DOI 10.1016/j.bulsci.2011.12.004
- Mouhamed Moustapha Fall and Enrico Valdinoci, Uniqueness and nondegeneracy of positive solutions of $(-\Delta )^su+u=u^p$ in $\Bbb R^N$ when $s$ is close to 1, Comm. Math. Phys. 329 (2014), no. 1, 383–404. MR 3207007, DOI 10.1007/s00220-014-1919-y
- Rupert L. Frank and Enno Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $\Bbb {R}$, Acta Math. 210 (2013), no. 2, 261–318. MR 3070568, DOI 10.1007/s11511-013-0095-9
- Rupert L. Frank, Enno Lenzmann, and Luis Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math. 69 (2016), no. 9, 1671–1726. MR 3530361, DOI 10.1002/cpa.21591
- Ailana Fraser and Richard Schoen, Sharp eigenvalue bounds and minimal surfaces in the ball, Invent. Math. 203 (2016), no. 3, 823–890. MR 3461367, DOI 10.1007/s00222-015-0604-x
- Vincent Millot and Yannick Sire, On a fractional Ginzburg-Landau equation and 1/2-harmonic maps into spheres, Arch. Ration. Mech. Anal. 215 (2015), no. 1, 125–210. MR 3296146, DOI 10.1007/s00205-014-0776-3
- Armin Schikorra, Regularity of $n/2$-harmonic maps into spheres, J. Differential Equations 252 (2012), no. 2, 1862–1911. MR 2853564, DOI 10.1016/j.jde.2011.08.021
Additional Information
- Yannick Sire
- Affiliation: Department of Mathematics, Johns Hopkins University, 404 Krieger Hall, 3400 N. Charles Street, Baltimore, Maryland 21218
- MR Author ID: 734674
- Email: sire@math.jhu.edu
- Juncheng Wei
- Affiliation: Department of Mathematics, University of British Columbia, Vancouver V6T 1Z2, Canada
- MR Author ID: 339847
- ORCID: 0000-0001-5262-477X
- Email: jcwei@math.ubc.ca
- Youquan Zheng
- Affiliation: School of Mathematics, Tianjin University, Tianjin 300072, People’s Republic of China
- Email: zhengyq@tju.edu.cn
- Received by editor(s): December 20, 2017
- Received by editor(s) in revised form: April 5, 2018
- Published electronically: September 17, 2018
- Additional Notes: The third author was partially supported by NSF of China (11301374) and China Scholarship Council (CSC)
- Communicated by: Joachim Krieger
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 5263-5268
- MSC (2010): Primary 35B06
- DOI: https://doi.org/10.1090/proc/14184
- MathSciNet review: 3866865