a positive weight. Another approach is presented in [@MRL], where boundary datum belongs to $L^1(\partial\Omega)$, but the boundary condition is understood in a weaker sense.

Throughout this paper $\Omega \subset \mathbb{R}^N$ shall be an open, bounded, strictly convex set with Lipschitz (or $C^1$) boundary. The boundary datum $f$ will belong to $L^1(\partial \Omega)$ or $BV(\partial\Omega)$. We consider the following minimalization problem called the least gradient problem (for brevity denoted by LGP):

$$\label{zagadnienie}
\inf \{ \int_\Omega |Du|, \quad u \in BV(\Omega), \quad Tu = f \},$$

where $T$ denotes the trace operator $T: BV(\Omega) \rightarrow L^1(\partial\Omega)$. Even existence of solutions in this sense is not obvious, as the functional

$$F(u) = \twopartdef{\int_\Omega |Du|}{ u \in BV(\Omega) \text{ and } Tu = f;}{+\infty}{\text{otherwise}}$$

is not lower semicontinuous with respect to $L^1$ convergence. In fact, in [@ST] the authors have given an example of a function $f$ without a solution to corresponding least gradient problem. It was a characteristic function of a certain fat Cantor set. Let us note that it does not lie in $BV(\partial\Omega)$.

There are two possible ways to deal with Problem . The first is the relaxation of the functional $F$. Such reformulation and its relationship with the original 