On the regularity of reflector antennas

where ν is the outer normal to A at the point where the light ray hits A. Suppose that we have a light source located at O, and Ω,Ω∗ are two domains in the sphere Sn−1, f(x) is a positive function for x ∈ Ω (input illumination intensity), and g(x∗) is a positive function for x∗ ∈ Ω∗ (output illumination intensity). If light emanates from O with intensity f(x) for x ∈ Ω, the far field reflector antenna problem is to find a perfectly reflecting surface A parametrized by z = ρ(x)x for x ∈ Ω, such that all reflected rays by A fall in the directions in Ω∗, and the output illumination received in the direction x∗ is g(x∗); that is, T (Ω) = Ω∗, where T is given by (1.1). Assuming there is no loss of energy in the reflection, then by the law of conservation of energy ∫