Pointwise estimates and regularity in geometric optics and other Generated Jacobian Equations

TL;DR

This paper develops pointwise estimates for weak solutions of generated Jacobian equations in geometric optics, advancing the regularity theory beyond optimal transport and addressing the near field reflector problem.

Contribution

It introduces new pointwise estimates for solutions of generated Jacobian equations, extending regularity results in geometric optics beyond optimal transport frameworks.

Findings

Established pointwise estimates under A3w-like conditions

Extended regularity theory to generated Jacobian equations

Addressed the near field reflector problem

Abstract

The study of reflector surfaces in geometric optics necessitates the analysis of certain nonlinear equations of Monge-Amp\`ere type known as generated Jacobian equations. These equations, whose general existence theory has been recently developed by Trudinger go beyond the framework of optimal transport. We obtain pointwise estimates for weak solutions of such equations under a condition analogous to the A3w condition of Ma, Trudinger and Wang. Estimates of this type have played an important role in the regularity theory for optimal transport maps and were previously unknown in this context, including the important case of the near field reflector problem.