Startseite Regularity and quantification for harmonic maps with free boundary
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

Regularity and quantification for harmonic maps with free boundary

  • Paul Laurain und Romain Petrides EMAIL logo
Veröffentlicht/Copyright: 4. November 2015

Abstract

We prove a quantification result for harmonic maps with free boundary and finite energy from arbitrary Riemannian surfaces into the unit ball of n+1. This generalizes results obtained by Da Lio [1] on the disk.

MSC 2010: 58J05

Communicated by Frank Duzaar


Acknowledgements

The first author was visiting the Department of Mathematics of Stanford University when this article was written and he would like to express his thanks for the hospitality and the excellent working conditions.

References

[1] Da Lio F., Compactness and bubbles analysis for 1/2-harmonic maps, Ann. Inst. H. Poincaré Anal. Non Linéaire 32 (2015), no. 1, 201–224. 10.1016/j.anihpc.2013.11.003Suche in Google Scholar

[2] Da Lio F. and Rivière T., Three-term commutator estimates and the regularity of 12-harmonic maps into spheres, Anal. PDE 4 (2011), no. 1, 149–190. 10.2140/apde.2011.4.149Suche in Google Scholar

[3] Duzaar F. and Steffen K., A partial regularity theorem for harmonic maps at a free boundary, Asymptot. Anal. 2 (1989), no. 4, 299–343. 10.3233/ASY-1989-2403Suche in Google Scholar

[4] Fraser A. and Schoen R., Minimal surfaces and eigenvalue problems, Geometric Analysis, Mathematical Relativity, and Nonlinear Partial Differential Equations, Contemp. Math. 599, American Mathematical Society, Providence (2013), 105–121. 10.1090/conm/599/11927Suche in Google Scholar

[5] Fraser A. and Schoen R., Uniqueness theorems for free boundary minimal disks in space forms, Int. Math. Res. Not. IMRN 2015 (2015), no. 17, 8268–8274. 10.1093/imrn/rnu192Suche in Google Scholar

[6] Hélein F., Harmonic Maps, Conservation Laws and Moving Frames, 2nd ed., Cambridge Tracts in Math. 150, Cambridge University Press, Cambridge, 2002. 10.1017/CBO9780511543036Suche in Google Scholar

[7] Laurain P. and Rivière T., Angular energy quantization for linear elliptic systems with antisymmetric potentials and applications, Anal. PDE 7 (2014), no. 1, 1–41. 10.2140/apde.2014.7.1Suche in Google Scholar

[8] Parker T. H., Bubble tree convergence for harmonic maps, J. Differential Geom. 44 (1996), no. 3, 595–633. 10.4310/jdg/1214459224Suche in Google Scholar

[9] Petrides R., Existence and regularity of maximal metrics for the first Laplace eigenvalue on surfaces, Geom. Funct. Anal. 24 (2014), no. 4, 1336–1376. 10.1007/s00039-014-0292-5Suche in Google Scholar

[10] Petrides R., Maximizing Steklov eigenvalues on surfaces, preprint 2015. 10.4310/jdg/1567216955Suche in Google Scholar

[11] Rivière T., Conformally invariant variational problems, preprint 2012, http://arxiv.org/abs/1206.2116. Suche in Google Scholar

[12] Sacks J. and Uhlenbeck K., The existence of minimal immersions of 2-spheres, Ann. of Math. (2) 113 (1981), no. 1, 1–24. 10.2307/1971131Suche in Google Scholar

[13] Scheven C., Partial regularity for stationary harmonic maps at a free boundary, Math. Z. 253 (2006), no. 1, 135–157. 10.1007/s00209-005-0891-9Suche in Google Scholar

Received: 2015-6-2
Revised: 2015-9-14
Accepted: 2015-9-22
Published Online: 2015-11-4
Published in Print: 2017-1-1

© 2017 by De Gruyter

Heruntergeladen am 10.9.2025 von https://www.degruyterbrill.com/document/doi/10.1515/acv-2015-0026/html
Button zum nach oben scrollen