Regularity for the near field parallel refractor and reflector problems

TL;DR

This paper establishes local regularity estimates for solutions to the near field parallel refractor and reflector problems, focusing on target set conditions without requiring smooth densities.

Contribution

It provides new local $C^{1,eta}$ regularity results for solutions under minimal assumptions on the target set and densities.

Findings

Proves local $C^{1,eta}$ regularity of solutions.

Applies to problems with non-smooth densities.

Works under minimal assumptions on the target set.

Abstract

We prove local $C^{1,\alpha}$ estimates of solutions for the parallel refractor and reflector problems under local assumptions on the target set $\Sigma$, and no assumptions are made on the smoothness of the densities.