On the Dirichlet problem for Monge-Ampere type equations

TL;DR

This paper establishes second derivative estimates and classical solutions for the Dirichlet problem of Monge-Ampere type equations under sharp conditions, advancing understanding in optimal transportation and prescribed Jacobian equations.

Contribution

It provides new second derivative estimates and solvability results for Monge-Ampere type equations with specific regularity and subsolution conditions.

Findings

Proved second derivative estimates for the equations.

Established classical solvability under sharp hypotheses.

Applied results to optimal transportation and Jacobian equations.

Abstract

In this paper, we prove second derivative estimates together with classical solvability for the Dirichlet problem of certain Monge-Ampere type equations under sharp hypotheses. In particular we assume that the matrix function in the augmented Hessian is regular in the sense used by Trudinger and Wang in their study of global regularity in optimal transportation [28] as well as the existence of a smooth subsolution. The latter hypothesis replaces a barrier condition also used in their work. The applications to optimal transportation and prescribed Jacobian equations are also indicated.