Sobolev spaces of maps and the Dirichlet problem for harmonic maps

TL;DR

This paper proves the existence of solutions to the Dirichlet problem for harmonic maps into convex geodesic balls without curvature restrictions, using Sobolev space analysis and deformation techniques.

Contribution

It improves a classical theorem by removing curvature assumptions and introduces new analysis methods for Sobolev spaces of maps.

Findings

Existence of harmonic map solutions into convex geodesic balls.

No curvature restrictions needed on the target.

Development of Sobolev space analysis for harmonic maps.

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.