Transport Energy

TL;DR

This paper introduces a new energy functional related to optimal transport, studies its gradient flow, and demonstrates convergence of approximations to the optimal transport density through PDE analysis.

Contribution

It defines a novel transport energy functional, analyzes its gradient flow, and establishes convergence of convex approximations to the optimal transport solution.

Findings

The unique minimizer of the transport energy is the optimal transport density.

Gradient flow of the energy converges to the optimal transport density.

Convex approximations converge to the true solution as parameters tend to zero.

Abstract

We introduce the \emph{transport energy} functional $\mathcal E$ (a variant of the Bouchitt\'e-Buttazzo-Seppecher shape optimization functional) and we prove that its unique minimizer is the optimal transport density $\mu^*$, i.e., the solution of Monge-Kantorovich equations. We study the gradient flow of $\mathcal E$ showing that $\mu^*$ is the unique global attractor of the flow. We introduce a two parameter family $\{\mathcal E_{\lambda,\delta}\}_{\lambda,\delta>0}$ of strictly convex functionals approximating $\mathcal E$ and we prove the convergence of the minimizers $\mu_{\lambda,\delta}^*$ of $\mathcal E_{\lambda,\delta}$ to $\mu^*$ as we let $\delta\to 0^+$ and $\lambda\to 0^+.$ We derive an evolution system of fully non-linear PDEs as gradient flow of $\mathcal E_{\lambda,\delta}$ in $L^2$, showing existence and uniqueness of solutions. All the trajectories of the flow converge in $W^{1,p}_0$ to the unique minimizer $\mu_{\lambda,\delta}^*$ of $\mathcal E_{\lambda,\delta}.$ Finally, we characterize $\mu_{\lambda,\delta}^*$ by a non-linear system of PDEs which is a perturbation of Monge-Kantorovich equations by means of a p-Laplacian.