Calculus of Variations and Geometric Measure Theory

R. Korte - P. Lahti - X. Li - N. Shanmugalingam

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

created by shanmugal on 08 Jan 2017
modified on 19 Jun 2018


Accepted Paper

Inserted: 8 jan 2017
Last Updated: 19 jun 2018

Journal: Revista Mat. Iberoamericana
Year: 2017


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