On the local theory of prescribed Jacobian equations revisited

Mathematical Sciences Institute, Australian National University, Canberra, Act 0200, Australia ,Neil S Trudinger

doi:10.3934/mine.2021048

Mathematical Sciences Institute, Australian National University, Canberra, Act 0200, Australia , Neil S Trudinger

Open Access

https://doi.org/10.3934/mine.2021048

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Abstract

<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. </abstract>

Highlights

Let Ω be a domain in Euclidean n-space, Rn, and Y a mapping from Ω × R × Rn into Rn
Det DY (·, u, Du) = ψ(·, u, Du), where ψ is a given scalar function on Ω × R × Rn and Du denotes the gradient of the function u : Ω → R
Denoting points in Ω × R × Rn by (x, z, p), we see that the special case, (1.2)

Summary

Introduction

Let Ω be a domain in Euclidean n-space, Rn , and Y a mapping from Ω × R × Rn into Rn. In particular we prove that the local convexity of smooth functions implies their global convexity for appropriately convex domains, that normal mappings are determined by sub-differentials and that sections and contact sets are convex in the extended sense. As well we prove that domain convexity with respect to generating functions is a special case of that determined by vector fields in [24, 25]. We remark that the existence of globally smooth elliptic solutions under corresponding conditions, including stronger domain convexity conditions, follows from the theory of the general prescribed Jacobian equation in [25] and its adaptation to near field reflection problems in [20].

Dpl Grs

Gyi Gz in Condition