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We study two notions of Dirichlet problem associated with BV energyminimizers (also called functions of least gradient) in bounded domains inmetric measure spaces whose measure is doubling and supports a$(1,1)$-Poincar\'e inequality. Since one of the two notions is not amenable tothe direct method of the calculus of variations, we construct, based on anapproach of [23, 29], solutions by considering the Dirichlet problem for$p$-harmonic functions, $p>1$, and letting $p\to 1$. Tools developed and usedin this paper include the inner perimeter measure of a domain.

Notions of Dirichlet problem for functions of least gradient in metric measure spaces