Refractor surfaces determined by near-field data

TL;DR

This paper investigates the near-field refractor problem with a point source and a prescribed target surface, establishing conditions for the local smoothness of solutions through Monge-Ampère equations and regularity criteria.

Contribution

It introduces a generating function framework for the refractor problem, proves the equivalence of Aleksandrov and Brenier solutions, and establishes local regularity results under smooth data and specific refractive index conditions.

Findings

Established local $C^2$ regularity of solutions with smooth data.

Derived Monge-Ampère type equations for the refractor problem.

Validated the MTW condition for a broad class of receiver surfaces.

Abstract

In this paper we study the near-field refractor problem with point source at the origin and prescribed target on the given receiver surface $\Sigma$. This nonvariational problem can be studied in the framework of prescribed Jacobian equations. We construct the corresponding generating function and show that the Aleksandrov and the Brenier type solutions are equivalent. Our main result establishes local smoothness of Aleksandrov's solutions when the data is smooth and when the medium containing the source has smaller refractive index than the medium containing the target. This is done by deriving the Monge-Ampere type equation that smooth solutions satisfy and establishing the validity of the MTW condition for a large class of receiver surfaces, which in turn implies the local $C^2 $ regularity of the refactor.