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Revista Matemática Iberoamericana

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Volume 35, Issue 6, 2019, pp. 1603–1648
DOI: 10.4171/rmi/1095

Published online: 2019-07-22

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

Riikka Korte[1], Panu Lahti[2], Xining Li[3] and Nageswari Shanmugalingam[4]

(1) Aalto University, Finland
(2) University of Cincinnati, USA and University of Jyväskylä, Finland
(3) Sun Yat-Sen University, Guangzhou, China
(4) University of Cincinnati, USA

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)-Poincaré 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 Mazón-Rossi–De León, 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: Function of bounded variation, inner trace, perimeter, least gradient, $p$-harmonic, Dirichlet problem, metric measure space, Poincaré inequality, codimension 1 Hausdorff measure

Korte Riikka, Lahti Panu, Li Xining, Shanmugalingam Nageswari: Notions of Dirichlet problem for functions of least gradient in metric measure spaces. Rev. Mat. Iberoam. 35 (2019), 1603-1648. doi: 10.4171/rmi/1095