On the local theory of prescribed Jacobian equations revisited

TL;DR

This paper revisits the local theory of prescribed Jacobian equations linked to generating functions, relaxing key conditions and extending regularity results using recent advances in convexity theory.

Contribution

It relaxes the monotonicity condition A4w and extends classical regularity theory to the weak form A3w of convexity conditions in prescribed Jacobian equations.

Findings

Relaxation of the A4w monotonicity condition.

Extension of regularity theory to the weak A3w condition.

Application of recent convexity results to Jacobian equations.

Abstract

In this paper we revisit our previous study of the local theory of prescribed Jacobian equations associated with generating functions, which are extensions of cost functions in the theory of optimal transportation. In particular, as foreshadowed in the earlier work, we provide details pertaining to the relaxation of a monotonicity condition A4w in the underlying convexity theory and the consequent classical regularity. Taking advantage of recent work of Kitagawa and Guillen, we also extend our classical regularity theory to the weak form A3w of the critical matrix convexity conditions.