On the design of a reflector antenna

We consider the problem of recovering a reflecting surface such that for a given point source of light the directions of the reflected rays cover a prescribed region of a far field sphere and the density of the distribution of the reflected rays is a function prescribed in advance, where the aperture of the incident ray cone is also prescribed in advance. Mathematically this problem requires one to solve a nonlinear partial differential equation of Monge - Ampère type subject to a nonlinear boundary condition. Numerical computations for the problem have been carried out by several authors. In this paper we study the existence, uniqueness, and smoothness of the reflecting surfaces for the above problem.