The planar Least Gradient problem in convex domains: the discontinuous case

TL;DR

This paper investigates the existence of solutions to the two-dimensional least gradient problem in convex polygonal domains with possibly discontinuous boundary data, extending classical results to non-strictly convex settings.

Contribution

It establishes existence results for the least gradient problem with discontinuous boundary data in convex polygons, introducing admissibility conditions and a limiting construction method.

Findings

Solutions exist when boundary data are in BV and satisfy admissibility conditions.

Classical results do not apply due to lack of strict convexity and discontinuities.

A limiting process constructs solutions from known problems.

Abstract

We study the two dimensional least gradient problem in convex polygonal sets in the plane, $\Omega$. We show the existence of solutions when the boundary data $f$ are attained in the trace sense. The main difficulty here is a possible discontinuity of $f$. Moreover, due to the lack of strict convexity of $\Omega$, the classical results are not applicable. We state the admissibility conditions on the boundary datum $f$, that are sufficient for establishing an existence result. One of them is that $f\in BV(\partial\Omega)$. The solutions are constructed by a limiting process, which uses solutions to known problems