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

Abstract

We introduce and study a new optimal transport problem on a bounded domain Ω ⊂ 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 Ω on the one hand, and on the other hand between two independent optimal transport problems set on Ω and ∂Ω.

Dynamic optimal transport on networks

“…having support on edges and vertices and allow for transport along the edges which are coupled to the vertices via the formal outflux. For each edge, this translates to an unbalanced transport problem as in [16]. After the formulation of the problem, we show that it is well-defined again using Fenchel-Rockafellar duality, see Theorem 4.9.…”

“…Kantorovich formulation and limit problem Another interesting question for further research is a static formulation of both (2) and (5). For the first one, based on the explicit calculation of geodesics in [16] between two point masses, one being located within the domain and one at the boundary, we conjecture that (2) allows for a Kantorovich formulation with cost…”

“…The goal of this section is to extend the model of [16] to multiple coupled equations on a network. We focus on the two-dimensional situation and a planar network where edges can be identified with one-dimensional domains.…”

“…In this section, we proof existence of minimisers of (5), based on duality, by extending the strategy of [16] to the network setting.…”

“…More recently, Monsaingeon (in [16]) introduced a new transportation model on the closure of a domain Ω that behaves differently in the interior and on the boundary while allowing for interaction between these two. This setting can be motivated if we think of Ω as a city with a ring road ∂Ω where cars have to pay a toll, that we denote by κ, if they enter the ring road.…”