On the local theory of prescribed Jacobian equations revisited

• Main Author: Neil S. Trudinger
• Format: Article
• Language: English
• Published: AIMS Press 2021-03-01
• Series: Mathematics in Engineering
• Subjects: prescribed jacobian equations; generating functions; convexity theory; existence and regularity
• Online Access: http://www.aimspress.com/article/doi/10.3934/mine.2021048?viewType=HTML

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 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.