The non-convex planar Least Gradient Problem

TL;DR

This paper investigates the existence of solutions to the least gradient problem in non-convex planar regions, utilizing optimal transportation theory to establish conditions under which solutions exist with continuous boundary data.

Contribution

It introduces new conditions for solution existence in non-convex regions, linking the least gradient problem to the Beckman problem and optimal transportation methods.

Findings

Established existence conditions for solutions in non-convex regions

Connected the least gradient problem to the Beckman problem

Applied optimal transportation theory to boundary value problems

Abstract

We study the least gradient problem in bounded regions with Lipschitz boundary in the plane. We provide a set of conditions for the existence of solutions in non-convex simply connected regions. We assume the boundary data is continuous and in the space of functions of bounded variation, and we are interested in solutions that satisfy the boundary conditions in the trace sense. Our method relies on the equivalence of the least gradient problem and the Beckman problem which allows us to use the tools of the optimal transportation theory.