On the second boundary value problem for Monge-Ampere type equations and geometric optics

TL;DR

This paper establishes the existence of classical solutions for second boundary value problems related to Monge-Ampère type equations, extending classical optimal transportation results to near field geometric optics using advanced derivative estimates and convexity theory.

Contribution

It introduces new existence results for generated prescribed Jacobian equations in geometric optics, leveraging second derivative estimates and convexity theory to avoid domain deformations.

Findings

Proved existence of classical solutions under weak convexity conditions.

Extended optimal transportation theory to geometric optics applications.

Utilized degree theory and convexity to construct solutions without domain deformation.

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.