  We show existence of solutions to the least gradient problem on the plane for
boundary data in $BV(\partial\Omega)$. We also provide an example of a function
$f \in L^1(\partial\Omega) \backslash (C(\partial\Omega) \cup
BV(\partial\Omega))$, 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.
