$p$-harmonic mappings between metric spaces

TL;DR

This paper extends the theory of $p$-harmonic maps to singular metric spaces, solving the Dirichlet problem for Sobolev maps and developing a trace theory for metric-valued Sobolev functions.

Contribution

It introduces a new approach to the Dirichlet problem for Sobolev maps between singular metric spaces and advances the trace theory for metric-valued Sobolev maps.

Findings

Extended Dirichlet problem solutions to singular metric spaces.

Developed a trace theory for Sobolev maps with metric values.

Characterized conditions for the existence of traces in borderline cases.

Abstract

In this paper, we solve the Dirichlet problem for Sobolev maps between singular metric spaces that extends the corresponding result of Guo and Wenger [Comm. Anal. Geom. 2020]. The main new ingredient in our proofs is a suitable extension of the theory of trace for metric valued Sobolev maps developed by Korevaar and Schoen [Comm. Anal. Geom. 1993]. We also develop a theory of trace in the borderline case, which investigates a sharp condition to characterize the existence of traces.