Notions of Dirichlet problem for functions of least gradient in metric measure spaces

TL;DR

This paper investigates two different notions of the Dirichlet problem for functions of least gradient in metric measure spaces, using p-harmonic functions as an approximation method to address challenges in direct variational approaches.

Contribution

It introduces a novel approach to solving the Dirichlet problem for BV energy minimizers in metric spaces by employing p-harmonic functions and perimeter measures.

Findings

Established existence of solutions via p-harmonic approximation

Developed tools involving inner perimeter measures

Extended Dirichlet problem concepts to metric measure spaces

Abstract

We study two notions of Dirichlet problem associated with BV energy minimizers (also called functions of least gradient) in bounded domains in metric measure spaces whose measure is doubling and supports a $(1,1)$-Poincar\'e inequality. Since one of the two notions is not amenable to the direct method of the calculus of variations, we construct, based on an approach of [23, 29], solutions by considering the Dirichlet problem for $p$-harmonic functions, $p>1$, and letting $p\to 1$. Tools developed and used in this paper include the inner perimeter measure of a domain.