The planar least gradient problem in convex domains: the discontinuous case

Abstract

We study the two dimensional least gradient problem in convex polygonal sets in the plane, Ω. 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 Ω , 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∈ BV(∂Ω). The solutions are constructed by a limiting process, which uses solutions to known problems. © 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG part of Springer Nature.