The planar Least Gradient problem in convex domains

TL;DR

This paper investigates the existence of solutions to the two-dimensional least gradient problem within convex domains, allowing for non-strict convexity and boundary data in BV space, under specific admissibility conditions.

Contribution

It establishes sufficient conditions on the boundary data and domain geometry that guarantee the existence of solutions to the least gradient problem in convex, not necessarily strictly convex, regions.

Findings

Existence of solutions under admissibility conditions

Extension to non-strictly convex domains

Solutions in BV function space with BV boundary data

Abstract

We study the two dimensional least gradient problem in a convex, but not necessary strictly convex region. We look for solutions in the space of $BV$ functions satisfying the boundary data $f$ in trace sense. We assume that $f$ is in $BV$ too. We state admissibility conditions on the trace and on the domain that are sufficient for existence of solutions.