An Analog of the Neumann Problem for the $1$-Laplace Equation in the Metric Setting: Existence, Boundary Regularity, and Stability

TL;DR

This paper investigates the existence, boundary regularity, and stability of solutions to a Neumann problem for the 1-Laplace equation in metric spaces, revealing conditions for regularity and nonuniqueness.

Contribution

It extends the theory of least gradient problems to metric spaces, establishing existence, boundary regularity, and stability results for solutions under various conditions.

Findings

Solutions exist under certain domain regularity assumptions.

Solutions can be expressed as differences of characteristic functions.

Solutions are boundary regular when the boundary has positive mean curvature.

Abstract

We study an inhomogeneous Neumann boundary value problem for functions of least gradient on bounded domains in metric spaces that are equipped with a doubling measure and support a Poincar\'e inequality. We show that solutions exist under certain regularity assumptions on the domain, but are generally nonunique. We also show that solutions can be taken to be differences of two characteristic functions, and that they are regular up to the boundary when the boundary is of positive mean curvature. By regular up to the boundary we mean that if the boundary data is $1$ in a neighborhood of a point on the boundary of the domain, then the solution is $-1$ in the intersection of the domain with a possibly smaller neighborhood of that point. Finally, we consider the stability of solutions with respect to boundary data.