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Pointwise Estimates and Regularity in Geometric Optics and Other Generated Jacobian Equations
Author(s) -
Guillen Nestor,
Kitagawa Jun
Publication year - 2017
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21691
Subject(s) - mathematics , pointwise , jacobian matrix and determinant , reflector (photography) , geometrical optics , type (biology) , nonlinear system , mathematical analysis , minkowski space , geometry , optics , physics , light source , ecology , quantum mechanics , biology
The study of reflector surfaces in geometric optics necessitates the analysis of certain nonlinear equations of Monge‐Ampère type known as generated Jacobian equations. This class of equations, whose general existence theory has been recently developed by Trudinger, goes beyond the framework of optimal transport. We obtain pointwise estimates for weak solutions of such equations under minimal structural and regularity assumptions, covering situations analogous to those of costs satisfying the A3‐weak condition introduced by Ma, Trudinger, and Wang in optimal transport. These estimates are used to develop a C 1,α regularity theory for weak solutions of Aleksandrov type. The results are new even for all known near‐field reflector/refractor models, including the point source and parallel beam reflectors, and are applicable to problems in other areas of geometry, such as the generalized Minkowski problem.© 2017 Wiley Periodicals, Inc.