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

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