Vol. 3, No. 1, 2021

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Local Hölder regularity of solutions to generated Jacobian equations

Seonghyeon Jeong

Vol. 3 (2021), No. 1, 163–188
Abstract

Generated Jacobian equations are Monge–Ampère type equations which contain optimal transport as a special case. Therefore, the optimal transport case has its own special structure, which is not necessarily true for more general generated Jacobian equations. Hence, the theory for optimal transport cannot be directly applied to generated Jacobian equations. In this paper, we point out the difficulties that prevent applying the proof of the local Hölder regularity of solutions of optimal transport problem from Loeper (2009) directly to generated Jacobian equations. We then discuss how to handle these difficulties and prove local Hölder regularity in the generated Jacobian equation case.

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Keywords
generated Jacobian equation, partial differential equation, elliptic partial differential equation, fully nonlinear partial differential equation
Mathematical Subject Classification
Primary: 35J60, 35J96
Milestones
Received: 26 May 2020
Revised: 12 December 2020
Accepted: 23 February 2021
Published: 28 May 2021
Authors
Seonghyeon Jeong
Department of Mathematics
Michigan State University
East Lansing, MI
United States