Notions of Dirichlet problem for functions of least gradient in 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.