Planar least gradient problem: existence, regularity and anisotropic case

TL;DR

This paper investigates the least gradient problem in the plane, establishing existence of solutions for boundary data in BV, demonstrating non-uniqueness in anisotropic cases, and extending regularity results to higher dimensions.

Contribution

It proves existence of solutions for BV boundary data, provides an example with less regular data, and analyzes non-uniqueness and regularity in anisotropic and higher-dimensional cases.

Findings

Existence of solutions for BV boundary data.

Non-uniqueness of solutions in anisotropic cases.

Regularity results applicable in higher dimensions.

Abstract

We show existence of solutions to the least gradient problem on the plane for boundary data in $BV(\partial\Omega)$. We also provide an example of a function $f \in L^1(\partial\Omega) \backslash (C(\partial\Omega) \cup BV(\partial\Omega))$, for which the solution exists. We also show non-uniqueness of solutions even for smooth boundary data in the anisotropic case for a nonsmooth anisotropy. We additionally prove a regularity result valid also in higher dimensions.