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

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.