Refractor surfaces determined by near-field data

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 Σ. 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-Ampère 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 C2 regularity of the refractor.