statement is considered in [@MRL] and [@Maz]. Another way is to consider when Problem has a solution in the classical sense and what is its regularity. This paper uses the latter approach.

The main result of the present paper is giving a sufficient condition for existence of solutions of the least gradient problem on the plane. It is given in the following theorem, which will be later proved as Theorem \[tw:istnienie\]:

Let $\Omega \subset \mathbb{R}^2$ be an open, bounded, strictly convex set with $C^1$ boundary. Then for every $f \in BV(\partial\Omega)$ there exists a solution of LGP for $f$.

Obviously, this condition is not necessary; the construction given in [@SWZ] does not require the boundary data to have finite total variation. We also provide an example of a function $f \in L^1(\Omega) \backslash (C(\partial\Omega) \cup BV(\partial\Omega))$, for which the solution exists, see Example \[ex:cantor\].

Another result included in this article provides a certain regularity property. Theorem \[tw:rozklad\] asserts existence of a decomposition of a function of least gradient into a continuous and a locally constant function. It is not a property shared by all BV functions, see [@AFP Example 4.1].

Let $\Omega \subset \mathbb{R}^N$, where $N \leq 7$, be an 