  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.
