Special cases of the planar least gradient problem

TL;DR

This paper investigates two specific instances of the planar least gradient problem, focusing on boundary conditions on parts of convex domains and on rectangles, analyzing solution existence and properties.

Contribution

It introduces new results on the existence and properties of solutions for least gradient problems in convex and non-strictly convex domains.

Findings

Existence of solutions in specified boundary conditions.

Properties of solutions depend on domain shape and data.

Analysis of particular data cases.

Abstract

We study two special cases of the planar least gradient problem. In the first one, the boundary conditions are imposed on a part of the strictly convex domain. In the second case, we impose the Dirichlet data on the boundary of a rectangle, an example of convex but not strictly convex domain. We show the existence of solutions and study their properties for particular cases of data.