$C^{1,\alpha}$ estimates for the parallel refractor

TL;DR

This paper establishes local regularity estimates for parallel refractors in media with different refractive indices, under geometric and regularity assumptions, and discusses conditions for their existence.

Contribution

It provides the first $C^{1,eta}$ regularity results for parallel refractors with higher refractive index media, under geometric and energy conservation assumptions.

Findings

Proved local $C^{1,eta}$ estimates for parallel refractors.

Established existence results under energy conservation.

Analyzed geometric conditions for regularity.

Abstract

We consider the parallel refractor problem when the planar radiating source lies in a medium having higher refractive index than the medium in which the target is located. We prove local $C^{1,\alpha}$ estimates for parallel refractors under suitable geometric assumptions on the source and target, and under local regularity hypotheses on the target set. We also discuss existence of refractors under energy conservation assumptions.