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, $Ω$. 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 $f\in BV(\partialΩ)$. The solutions are constructed by a limiting process, which uses solutions to known problems

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