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

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.