Title: On the second boundary value problem for Monge-Ampere type equations and geometric optics,
Abstract: In this paper, we prove the existence of classical solutions to second
boundary value prob- lems for generated prescribed Jacobian equations, as
recently developed by the second author, thereby obtaining extensions of
classical solvability of optimal transportation problems to problems arising in
near field geometric optics. Our results depend in particular on a priori
second derivative estimates recently established by the authors under weak
co-dimension one convexity hypotheses on the associated matrix functions with
respect to the gradient variables, (A3w). We also avoid domain deformations by
using the convexity theory of generating functions to construct unique initial
solutions for our homotopy family, thereby enabling application of the degree
theory for nonlinear oblique boundary value problems.