DCDS
On Pogorelov estimates for Monge-Ampère type equations
Jiakun Liu Neil S. Trudinger
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
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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