Sobolev spaces of maps and the Dirichlet problem for harmonic maps
Differential Geometry
2017-11-28 v2 Analysis of PDEs
Abstract
In this paper we prove the existence of a solution to the Dirichlet problem for harmonic maps into a geodesic ball on which the squared distance function from the origin is strictly convex. This improves a celebrated theorem obtained by S. Hildebrandt, H. Kaul and K. Widman in 1977. In particular no curvature assumptions on the target are required. Our proof relies on a careful analysis of the Sobolev spaces of maps involved in the variational process, and on a deformation result which permits to glue a suitable Euclidean end to the geodesic ball.
Keywords
Cite
@article{arxiv.1412.3429,
title = {Sobolev spaces of maps and the Dirichlet problem for harmonic maps},
author = {Stefano Pigola and Giona Veronelli},
journal= {arXiv preprint arXiv:1412.3429},
year = {2017}
}
Comments
24 pages. Corrected typos. Added references