Continuity of minimizers to weighted least gradient problems

TL;DR

This paper establishes the existence and regularity of continuous minimizers for weighted least gradient problems in higher dimensions, extending previous techniques and ensuring minimal surface level sets under certain geometric conditions.

Contribution

It extends the Sternberg-Williams-Ziemer technique to inhomogeneous weighted problems, providing continuous solutions and analyzing their minimal surface level sets in high dimensions.

Findings

Constructed continuous minimizers in dimensions n>=2

Level sets are minimal surfaces in a conformal metric

Solutions exist under geometric conditions on the domain

Abstract

We revisit the question of existence and regularity of minimizers to weighted least gradient problems on a fixed bounded domain, subject to a Dirichlet boundary condition, in the case where the boundary data is continuous and the weight function is C^2 and bounded away from zero. Under suitable geometric conditions on the domain in R^n we construct continuous solutions of the above variational problem in any dimension n>=2, by extending the Sternberg-Williams-Ziemer technique to this setting of inhomogeneous variations. We show that the level sets of the constructed minimizer are minimal surfaces in a conformal metric determined by the weight function. This results complements the approach of Jerrard, Moradifam and Nachman since it provides a continuous solution even in high dimensions where the possibility exists for level sets to develop singularities. The proof relies on an application of a strict maximum principle for sets with area-minimizing boundary established by Leon Simon.