Domains in metric measure spaces with boundary of positive mean curvature, and the Dirichlet problem for functions of least gradient

TL;DR

This paper extends the theory of the Dirichlet problem for functions of least gradient to non-smooth metric measure spaces with boundaries of positive mean curvature, proving existence but not always uniqueness or continuity of solutions.

Contribution

It introduces a notion of boundary of positive mean curvature in metric measure spaces and proves existence of solutions to the Dirichlet problem for least gradients in this setting.

Findings

Existence of solutions for the Dirichlet problem in non-smooth spaces with positive mean curvature boundaries.

Counterexamples showing failure of uniqueness and continuous solutions even with Lipschitz weights.

Extension of classical results to more general metric measure space settings.

Abstract

We study the geometry of domains in complete metric measure spaces equipped with a doubling measure supporting a $1$-Poincar\'e inequality. We propose a notion of \emph{domain with boundary of positive mean curvature} and prove that, for such domains, there is always a solution to the Dirichlet problem for least gradients with continuous boundary data. Here \emph{least gradient} is defined as minimizing total variation (in the sense of BV functions) and boundary conditions are satisfied in the sense that the \emph{boundary trace} of the solution exists and agrees with the given boundary data. This extends the result of Sternberg, Williams and Ziemer to the non-smooth setting. Via counterexamples we also show that uniqueness of solutions and existence of \emph{continuous} solutions can fail, even in the weighted Euclidean setting with Lipschitz weights.