On the second boundary value problem for Monge-Ampere type equations and optimal transportation

TL;DR

This paper investigates the existence of smooth solutions to the second boundary value problem for Monge-Ampere equations, with applications to regularity in optimal transportation, extending previous results to broader cost functions.

Contribution

It establishes global second derivative estimates for solutions under weak conditions on cost functions, advancing regularity theory in optimal transportation.

Findings

Proved interior regularity for a broad class of cost functions.

Extended regularity results to include quadratic cost and other examples.

Derived global estimates for second derivatives of solutions.

Abstract

This paper is concerned with the existence of globally smooth solutions for the second boundary value problem for Monge-Ampere equations and the application to regularity of potentials in optimal transportation. The cost functions satisfy a weak form of our condition A3, under which we proved interior regularity in a recent paper with Xi-nan Ma. Consequently they include the quadratic cost function case of Caffarelli and Urbas as well as the various examples in the earlier work. The approach is through the derivation of global estimates for second derivatives of solutions.