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

### Riikka Korte

Aalto University, Finland### Panu Lahti

University of Cincinnati, USA and University of Jyväskylä, Finland### Xining Li

Sun Yat-Sen University, Guangzhou, China### Nageswari Shanmugalingam

University of Cincinnati, USA

## 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)-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 $p→1$. Tools developed and used in this paper include the inner perimeter measure of a domain.

## Cite this article

Riikka Korte, Panu Lahti, Xining Li, Nageswari Shanmugalingam, Notions of Dirichlet problem for functions of least gradient in metric measure spaces. Rev. Mat. Iberoam. 35 (2019), no. 6, pp. 1603–1648

DOI 10.4171/RMI/1095