Accepted Paper

Inserted: 8 jan 2017
Last Updated: 19 jun 2018

Journal: Revista Mat. Iberoamericana
Year: 2017

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 Juutinen and Mazon-Rossi-DeLeon, 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.

Keywords: perimeter, Poincare inequality, BV, inner trace, functions of least gradient, codimenson $1$ Hausdorff measure

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