mean curvature and is not locally area-minimizing). The proof of existence is constructive and we shall briefly recall it. The main idea is reversing Theorem \[twierdzeniezbgg\] and constructing almost all level sets of the solution. According to the Lemma \[lem:jednoznacznoscnadpoziomic\] this uniquely determines the solution.

We fix the boundary data $g \in  C(\partial \Omega)$. By Tietze theorem it has an extension $G \in C(\mathbb{R}^n \backslash \Omega)$. We may also demand that $G \in BV(\mathbb{R}^n \backslash \overline{\Omega})$. Let $L_t =  (\mathbb{R}^n \backslash \Omega) \cap \{ G \geq t \}$. Since $G \in BV(\mathbb{R}^n \backslash \overline{\Omega})$, then for a.e. $t \in \mathbb{R}$ we have $P(L_t, \mathbb{R}^n \backslash \overline{\Omega}) < \infty$. Let $E_t$ be a set solving the following problems:

$$\label{sternbergminimalnadlugosc}
\min \{ P(E, \mathbb{R}^n): E \backslash \overline{\Omega} = L_t \backslash \overline{\Omega} \},$$

$$\max \{ |E|: E \text{ is a minimizer of \eqref{sternbergminimalnadlugosc}} \}.$$

Let us note that both of these problems have solutions; let $m \geq 0$ be the infimum in the first problem. Let $E_n$ be a sequence of sets such that $P(E_n, \Omega) \rightarrow m$. By compactness of unit ball in $BV(\Omega)$ and lower semicontinuity of the total variation we obtain $\chi_{E_{n_k}} \rightarrow \chi_E$, where

$$m \leq P(E, \Omega) \leq 