Accepted Paper

Inserted: 22 jun 2017
Last Updated: 13 oct 2018

Journal: Journal of Geometric Analysis
Year: 2018

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.

Tags: GeMeThNES

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