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

Riikka Korte; Panu Lahti; Xining Li; Nageswari Shanmugalingam

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

Analysis of PDEs 2016-12-20 v1 Metric Geometry

Authors: Riikka Korte , Panu Lahti , Xining Li , Nageswari Shanmugalingam

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 [23, 29], 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

Dirichlet problem functions of least gradient metric measure spaces BV energy minimizers p-harmonic functions inner perimeter measure Poincaré inequality

Cite

@article{arxiv.1612.06078,
  title  = {Notions of Dirichlet problem for functions of least gradient in metric measure spaces},
  author = {Riikka Korte and Panu Lahti and Xining Li and Nageswari Shanmugalingam},
  journal= {arXiv preprint arXiv:1612.06078},
  year   = {2016}
}

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

Abstract

Keywords

Cite

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