Least gradient problem with respect to a non-strictly convex norm

TL;DR

This paper investigates the planar least gradient problem with anisotropic norms, establishing existence of minimizers in strictly convex domains and analyzing how convexity modes affect uniqueness and regularity.

Contribution

It provides new insights into the existence, uniqueness, and regularity of minimizers for the least gradient problem under anisotropic norms, independent of the norm's regularity.

Findings

Existence of minimizers in strictly convex domains

Analysis of how convexity modes influence uniqueness

Insights into regularity conditions for minimizers

Abstract

We study the planar least gradient problem with respect to an anisotropic norm $\phi$ for continuous boundary data. We prove existence of minimizers for strictly convex domains $\Omega$. Furthermore, we inspect the issue of uniqueness and regularity of minimizers only in terms of the modes of convexity of $\phi$ and $\Omega$. The results are independent from the regularity of $\phi$.