On the homotopy Dirichlet problem for p-harmonic maps

TL;DR

This paper addresses the homotopy Dirichlet problem for p-harmonic maps, providing complete solutions for compact targets and new proofs for specific non-compact cases, with implications for regularity and uniqueness.

Contribution

It offers a comprehensive solution to the homotopy Dirichlet problem for p-harmonic maps, including new proofs and a periodization method for non-compact targets.

Findings

Complete solution for compact target manifolds.

New proof for rotationally symmetric or two-dimensional targets.

Introduction of a periodization procedure for non-compact targets.

Abstract

In this two papers we deal with the relative homotopy Dirichlet problem for p-harmonic maps from compact manifolds with boundary to manifolds of non-positive sectional curvature. Notably, we give a complete solution to the problem in case the target manifold is either compact and a new proof in case it is rotationally symmetric or two dimensional and simply connected. The proof of the compact case uses some ideas of White to define the relative d-homotopy type of Sobolev maps, and the regularity theory by Hardt and Lin. To deal with non-compact targets we introduce a periodization procedure which permits to reduce the problem to the previous one. Also, a general uniqueness result is given.