# Limits of density-constrained optimal transport

@article{Gladbach2019LimitsOD, title={Limits of density-constrained optimal transport}, author={Peter Gladbach and Eva Kopfer}, journal={arXiv: Analysis of PDEs}, year={2019} }

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.

## One Citation

Homogenisation of dynamical optimal transport on periodic graphs

- Computer Science, MathematicsArXiv
- 2021

A homogenisation result is derived from a Γ-convergence result for action functionals on curves of measures that describes the effective behaviour of the discrete problems in terms of a continuous optimal transport problem.

## References

SHOWING 1-10 OF 29 REFERENCES

A-QUASICONVEXITY: RELAXATION AND HOMOGENIZATION

- Mathematics
- 2000

Integral representation of relaxed energies and of Γ -limits of functionals are obtained when sequences of fields v may develop oscillations and are constrained to satisfy a system of first order…

THE GEOMETRY OF DISSIPATIVE EVOLUTION EQUATIONS: THE POROUS MEDIUM EQUATION

- Mathematics
- 2001

We show that the porous medium equation has a gradient flow structure which is both physically and mathematically natural. In order to convince the reader that it is mathematically natural, we show…

Transport Equations and Multi-D Hyperbolic Conservation Laws

- Mathematics
- 2008

I.- Existence, Uniqueness, Stability and Differentiability Properties of the Flow Associated to Weakly Differentiable Vector Fields.- II.- A Note on Alberti's Rank-One Theorem.- III.- Regularizing…

First Order Mean Field Games with Density Constraints: Pressure Equals Price

- Mathematics, Computer ScienceSIAM J. Control. Optim.
- 2016

A minimal regularity is obtained, which allows us to write optimality conditions at the level of single-agent trajectories and to define a weak notion of Nash equilibrium for the authors' model.

Euler sprays and Wasserstein geometry of the space of shapes

- Mathematics
- 2016

We study a distance between shapes defined by minimizing the integral of kinetic energy along transport paths constrained to measures with characteristic-function densities. The formal geodesic…

Optimal transportation with capacity constraints

- Mathematics
- 2012

The classical problem of optimal transportation can be formulated as a linear optimization problem on a convex domain: among all joint measures with fixed marginals find the optimal one, where…

Stochastic Differential Equations

- Mathematics
- 1998

We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion…

Least action principles for incompressible flows and optimal transport between shapes

- Mathematics
- 2016

As V. I. Arnold observed in the 1960s, the Euler equations of incompressible fluid flow correspond formally to geodesic equations in a group of volume-preserving diffeomorphisms. Working in an…

A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem

- Mathematics, Computer ScienceNumerische Mathematik
- 2000

Summary. The
$L^2$ Monge-Kantorovich mass transfer problem [31] is reset in a fluid mechanics framework and numerically solved by an augmented Lagrangian method.

Variational Mean Field Games

- Mathematics
- 2017

This paper is a brief presentation of those mean field games with congestion penalization which have a variational structure, starting from the deterministic dynamical framework. The stochastic…