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.