On the local theory of prescribed Jacobian equations

TL;DR

This paper develops a local regularity theory for prescribed Jacobian equations, extending optimal transportation results by incorporating a generating function with an additional scalar variable.

Contribution

It introduces a generalized local regularity framework for prescribed Jacobian equations, extending existing optimal transportation theories with new convexity and regularity results.

Findings

Established foundational regularity results for the generalized equations.

Extended the convexity theory to include generating functions with scalar dependence.

Unified the local theory for potentials of Ma, Trudinger, and Wang within this new framework.

Abstract

We develop the fundamentals of a local regularity theory for prescribed Jacobian equations which extend the corresponding results for optimal transportation equations. In this theory the cost function is extended to a generating function through dependence on an additional scalar variable. In particular we recover in this generality the local regularity theory for potentials of Ma,Trudinger and Wang, along with the subsequent development of the underlying convexity theory.