A parabolic flow toward solutions of the optimal transportation problem on domains with boundary

TL;DR

This paper introduces a parabolic flow approach to solve the optimal transportation problem on bounded domains, demonstrating convergence to the solution under specific conditions on the cost function and domains.

Contribution

It presents a novel parabolic flow method for the optimal transportation problem and proves its convergence to the solution under certain assumptions.

Findings

Convergence of the parabolic flow to the optimal transport solution.

Conditions on the cost function and domains ensure convergence.

Provides a new dynamic approach to solving mass transport problems.

Abstract

We consider a parabolic version of the mass transport problem, and show that it converges to a solution of the original mass transport problem under suitable conditions on the cost function, and initial and target domains.