On the design of reflectors with prespecified distribution of virtual sources and intensities

We consider the problem of determining a reflecting convex surface transforming the energy flow from a given point source of light into an energy flow from a prescribed-in-advance set of virtual sources. This set may consist of a finite set of point sources or of distributed sources. The problem is studied in the geometrical optics approximation. The analytic formulation of the problem leads to a complicated nonlinear partial differential equation of Monge-Ampère type. Here, we formulate the problem in terms of certain associated measures and prove the existence of weak solutions.