Planar least gradient problem: existence,regularity and anisotropic case

We show existence of solutions to the least gradient problem on the plane for boundary data in \(BV(\partial \varOmega )\). We also provide an example of a function \(f \in L^1(\partial \varOmega ) \backslash \) \((C(\partial \varOmega ) \cup BV(\partial \varOmega ))\), for which the solution exists. We also show non-uniqueness of solutions even for smooth boundary data in the anisotropic case for a nonsmooth anisotropy. We additionally prove a regularity result valid also in higher dimensions.