---
abstract: '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.'
address: 'W. Górny: Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Warsaw, Poland.'
author:
- Wojciech Górny
title: 'Planar least gradient problem: existence, regularity and anisotropic case'
---

Introduction
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Many papers, including [@SWZ], [@MNT], [@MRL], [@GRS] describe the least gradient problem, i.e. a problem of minimalization

$$\min \{ \int_\Omega |Du|, \quad u \in BV(\Omega), \quad u|_{\partial\Omega} = f  \},$$

where we may impose certain conditions on $\Omega$, $f$ and use different approaches to the boundary condition. In [@SWZ] $f$ is assumed to be continuous and the boundary condition is in the sense of traces. They also impose a set of geometrical conditions on $\Omega$, which are satisfied by strictly convex sets; in fact, in dimension two they are equivalent to strict convexity. The authors of [@MNT] also add 