On the local theory of prescribed Jacobian equations revisited

Neil S. Trudinger; Neil S. Trudinger

doi:10.3934/mine.2021048

Mathematics in Engineering

2021, Volume 3, Issue 6: 1-17. doi: 10.3934/mine.2021048

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On the local theory of prescribed Jacobian equations revisited

Neil S. Trudinger ^,

Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia

Received: 17 December 2020 Accepted: 14 January 2021 Published: 01 February 2021

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.
- prescribed Jacobian equations,
- generating functions,
- convexity theory,
- existence and regularity
Citation: Neil S. Trudinger. On the local theory of prescribed Jacobian equations revisited[J]. Mathematics in Engineering, 2021, 3(6): 1-17. doi: 10.3934/mine.2021048

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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 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.

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