The Dirichlet Problem for Orlicz-Sobolev mappings between metric space

TL;DR

This paper extends the theory of Orlicz-Sobolev mappings between metric spaces by solving the Dirichlet problem, developing compactness theorems, and extending trace theory, advancing the understanding of nonlinear analysis in singular metric contexts.

Contribution

It introduces new results on the Dirichlet problem for Orlicz-Sobolev maps between metric spaces, including compactness and trace extension theorems, generalizing previous work to broader settings.

Findings

Solved the Dirichlet problem for Orlicz-Sobolev maps between metric spaces.

Developed a Rellich-Kondrachov compactness theorem for these mappings.

Extended trace theory for metric-valued Sobolev maps.

Abstract

In this paper, we solve the Dirichlet problem for Orlicz-Sobolev maps between singular metric spaces that extends the corresponding result of Guo et al. [arXiv 2021]. As an intermediate step, we develop a version of Rellich-Kondrachov compactness theorem for Orlicz-Sobolev mappings between metric spaces that extends a previous result of Guo and Wenger [Comm. Anal. Geom. 2020]. Another crucial ingredient is an Orlicz-Sobolev extension of the trace theory for metric valued Sobolev maps developed by Korevaar and Schoen [Comm. Anal. Geom. 1993].