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

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.