The planar Least Gradient problem in convex domains, the case of continuous datum

TL;DR

This paper investigates the two-dimensional least gradient problem in convex polygons, establishing existence and uniqueness of solutions with continuous boundary data despite the lack of strict convexity.

Contribution

It introduces admissibility conditions for continuous boundary data in convex polygons, extending classical results to non-strictly convex domains.

Findings

Existence of solutions under new admissibility conditions

Solutions constructed via a limiting process using superlevel set geometry

Uniqueness of solutions in the specified setting

Abstract

We study the two dimensional least gradient problem in a convex polygonal set in the plane. We show existence of solutions when the boundary data are attained in the trace sense. Due to the lack of strict convexity, the classical results are not applicable. We state the admissibility conditions on the continuous boundary datum $f$ that are sufficient for establishing an existence and uniqueness result. The solutions are constructed by a limiting process, which uses the well-known geometry of superlevel sets of least gradient functions.