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By the Snell law of reflection, a light ray incident upon a reflective surface will be reflected at an angle equal to the incident angle. Both angles are measured with respect to the normal to the surface. If a light ray emanates from O in the direction x ∈ S n−1 , and A is a perfectly reflecting surface, then the reflected ray has direction: (1.1) x∗ = T (x) = x − 2 x, ν ν, where ν is the outer normal to A at the point where the light ray hits A. | Annals of Mathematics On the regularity of reflector antennas By Luis A. Caffarelli Cristian E. Guti errez and Qingbo Huang Annals of Mathematics 167 2008 299-323 On the regularity of reflector antennas By Luis A. Caffarelli Cristian E. Gutierrez and Qingbo Huang 1. Introduction By the Snell law of reflection a light ray incident upon a reflective surface will be reflected at an angle equal to the incident angle. Both angles are measured with respect to the normal to the surface. If a light ray emanates from O in the direction x E Sn-1 and A is a perfectly reflecting surface then the reflected ray has direction 1.1 x T x x 2 x V V where V 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 Q Q are two domains in the sphere Sn-1 f x is a positive function for x E Q input illumination intensity and g x is a positive function for x E Q output illumination intensity . If light emanates from O with intensity f x for x E Q the far field reflector antenna problem is to find a perfectly reflecting surface A parametrized by z p x x for x E Q such that all reflected rays by A fall in the directions in Q and the output illumination received in the direction x is g x that is T Q Q 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 y f x dx y g x dx . In addition and again by conservation of energy the map T defined by 1.1 is measure-preserving y f x dx y g x dx for all E c Q Borel set The first author was partially supported by NSF grant DMS-0140338. The second author was partially supported by NSF grant DMS-0300004. The third author was partially supported by NSF grant DMS-0201599. 300 L. A. CAFFARELLI C. E. GUTIERREZ AND Q. HUANG and consequently the Jacobian of T is ear equation on Sn-1 see GW98 f x g T x . It yields the following nonlin- 1-2 det Viju u - n eij _ f x nn 1 det eij g T x where u 1 p V covariant derivative n Vu 2 u2 2u and e
