A new transportation distance with bulk/interface interactions and flux penalization

We introduce and study a new optimal transport problem on a bounded domain $\bar{\mathrm{\Omega }}\subset {\mathbb{R}}^d$, defined via a dynamical Benamou-Brenier formulation. The model handles differently the motion in the interior and on the boundary, and penalizes the transfer of mass between the two. The resulting distance interpolates between classical optimal transport on $\bar{\mathrm{\Omega }}$ on the one hand, and on the other hand between two independent optimal transport problems set on $\mathrm{\Omega }$ and $\mathrm{\partial }\mathrm{\Omega }$.