On a Monge-Ampère equation arising in geometric optics (Q1265275)

A smooth convex surface \(F\) in \(\mathbb{R}^3\), star-shaped relative to a point \(O\), defines a ``mirror map'' if the point \(O\) is treated as a point-source of light and \(F\) is the mirror. The directions of reflected rays define a subset of a unit sphere (so called far-sphere). The inverse problem is to determine the surface \(F\) from the given point \(O\), input aperture, the image on the far-sphere, and the input and output intensity distributions. In analytic form this problem leads to an equation of Monge-Ampère type on a subdomain of a unit sphere. The authors show that for this equation one can derive \(C^2\) a priori estimates and, therefore, one can establish existence of smooth solutions.