An inverse problem for the refractive surfaces with parallel lighting

In this article we examine the regularity of two types of weak solutions to a Monge-Amp\`ere type equation which emerges in a problem of finding surfaces that refract coaxial light rays emitted from source domain and striking a given target after refraction. Historically, ellipsoids and hyperboloids of revolution were the first surfaces to be considered in this context. The mathematical formulation commences with deriving the energy conservation equation for sufficiently smooth surfaces, regarded as graphs of functions to be sought, and then studying the existence and regularity of two classes of suitable weak solutions constructed from envelopes of hyperboloids or ellipsoids of revolution. Our main result in this article states that under suitable conditions on source and target domains and respective intensities these weak solutions are locally smooth.