Cristian E. Gutiérrez: Refraction problems in geometric optics

Given two homogeneous and isotropic materials I and II with refractive indices nI and nII, the problems described in this talk concern with the design of interface surfaces between the two materials, so that radiation emanating from one source lying in medium I and having a given intensity depending of the direction, is refracted by the surface into a prescribed destination, such us a set of directions or a target lying in medium II, and with a given intensity, in general a Radon measure. This leads to the design of lenses refracting radiation in a prescribed manner. We prove existence of these surfaces using an optimization process when the destination is a finite set, and then by a limiting process in the general case. The surface solutions satisfy nonlinear pdes of Monge-Ampere type.