# Limits of density-constrained optimal transport

## TL;DR

This paper investigates the limits of density-constrained dynamic optimal transport, deriving variational limits for singular phenomena such as permeable membranes and homogenized porous media.

## Contribution

It introduces new variational limit results for optimal transport under density constraints, including membrane and homogenization effects.

## Key findings

- Optimal flow through an infinitesimal permeable membrane.
- Homogenized optimal flow through a porous medium.
- Gamma-convergence characterizes the limits of constrained optimal transport.

## Abstract

We consider the problem of dynamic optimal transport with a density constraint. We derive variational limits in terms of $\Gamma$-convergence for two singular phenomena. First, for densities constrained near a hyperplane we recover the optimal flow through an infinitesimal permeable membrane. Second, for rapidly oscillating periodic constraints we obtain the optimal flow through a homogenized porous medium.