DCDS
On Pogorelov estimates for Monge-Ampère type equations
Jiakun Liu Neil S. Trudinger
Discrete & Continuous Dynamical Systems - A 2010, 28(3): 1121-1135 doi: 10.3934/dcds.2010.28.1121
In this paper, we prove interior second derivative estimates of Pogorelov type for a general form of Monge-Ampère equation which includes the optimal transportation equation. The estimate extends that in a previous work with Xu-Jia Wang and assumes only that the matrix function in the equation is regular with respect to the gradient variables, that is it satisfies a weak form of the condition introduced previously by Ma,Trudinger and Wang for regularity of optimal transport mappings. We also indicate briefly an application to optimal transportation.
keywords: optimal transportation. Pogorelov estimates
DCDS
On the classical solvability of near field reflector problems
Jiakun Liu Neil S. Trudinger
Discrete & Continuous Dynamical Systems - A 2016, 36(2): 895-916 doi: 10.3934/dcds.2016.36.895
In this paper we prove the existence of classical solutions to near field reflector problems, both for a point light source and for a parallel light source, with planar receivers. These problems involve Monge-Ampère type equations, subject to nonlinear oblique boundary conditions. Our approach builds on earlier work in the optimal transportation case by Trudinger and Wang and makes use of a recent extension of degree theory to oblique boundary conditions by Li, Liu and Nguyen.
keywords: existence obliqueness Monge-Ampère regularity. Reflector

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