Regularity theory for surfaces in geometric optics and other Generated Jacobian Equations

PDE Seminar
Tuesday, October 20, 2015 - 3:05pm
1 hour (actually 50 minutes)
Skiles 006
University of Massachusetts at Amherst
The study of reflector surfaces in geometric optics necessitates the analysis of nonlinear equations of Monge-Ampere type. For many important examples (including the near field reflector problem), the equation no longer falls within the scope of optimal transport, but within the class of "Generated Jacobian equations" (GJEs). This class of equations was recently introduced by Trudinger, motivated by problems in geometric optics, however they appear in many others areas (e.g. variations of the Minkowski problem in convex geometry). Under natural assumptions, we prove Holder regularity for the gradient of weak solutions. The results are new in particular for the near-field point source reflector problem, but are applicable for a broad class of GJEs: those satisfying an analogue of the A3-weak condition introduced by Ma, Trudinger and Wang in optimal transport. Joint work with Jun Kitagawa.