Weighted Beckmann problem with boundary costs

We show that a solution to a variant of the Beckmann problem can be obtained by studying the limit of some weighted $p-$Laplacian problems. More precisely, we find a solution to the following minimization problem: \begin{equation*} \min \bigg \{\int _\Omega k \mathrm {d}|w| + \int _{\partial \Omega } g^- \mathrm {d}\nu ^- - \int _{\partial \Omega } g^+ \mathrm {d}\nu ^+ : w \in \mathcal {M}^d(\Omega ), \nu \in \mathcal {M}(\partial \Omega ), -\nabla \cdot w =f + \nu \bigg \}, \end{equation*} where $f, k$, and $g^\pm$ are given. In addition, we connect this problem to a formulation with Kantorovich potentials with Dirichlet boundary conditions.