Maximum and comparison principles for convex functions on the Heisenberg group

We prove estimates, similar in form to the classical Aleksandrov estimates, for a Monge-Ampere type operator on the Heisenberg group. A notion of normal mapping does not seem to be available in this context and the method of proof uses integration by parts and oscillation estimates that lead to the construction of an analogue of Monge-Ampere measures for convex functions in the Heisenberg group.Comment: The results in this paper and the ideas of their proofs have been presented in the following talks: Analysis Seminar, Temple U., October 2002; Fabes--Chiarenza Lectures at Siracusa, December 2002; Pan-American Conference, Santiago de Chile, January 2003; Analysis Seminar, U. of Bologna, March 2003; and Analysis Seminar, U. Texas at Austin, March 200

Regularity for the near field parallel refractor and reflector problems

We prove local $C^{1,\alpha}$ estimates of solutions for the parallel refractor and reflector problems under local assumptions on the target set $\Sigma$, and no assumptions are made on the smoothness of the densities.Comment: 32 pages, three figure

Uniform Refraction in Negative Refractive Index Materials

We study the problem of constructing an optical surface separating two homogeneous, isotropic media, one of which has a negative refractive index. In doing so, we develop a vector form of Snell's law, which is used to study surfaces possessing a certain uniform refraction property, both in the near and far field cases. In the near field problem, unlike the case when both materials have positive refractive index, we show that the resulting surfaces can be neither convex nor concave.Comment: 29 pages, 5 figure

The Near Field Refractor

We present an abstract method in the setting of compact metric spaces which is applied to solve a number of problems in geometric optics. In particular, we solve the one source near field refraction problem. That is, we construct surfaces separating two homogenous media with different refractive indices that refract radiation emanating from the origin into a target domain contained in an n-1 dimensional hypersurface. The input and output energy are prescribed. This implies the existence of lenses focusing radiation in a prescribed manner.Comment: 39 pages, 4 figures, Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire (to appear). Geometric optics, lens design, measure equations, Descartes ovals, Monge-Ampere type equation

On the second order derivatives of convex functions on the Heisenberg group

In the Euclidean setting the celebrated Aleksandrov-Busemann-Feller theorem states that convex functions are a.e. twice differentiable. In this paper we prove that a similar result holds in the Heisenberg group, by showing that every continuous H-convex function belongs to the class of functions whose second order horizontal distributional derivatives are Radon measures. Together with a recent result by Ambrosio and Magnani, this proves the existence a.e. of second order horizontal derivatives for the class of continuous H-convex functions in the Heisenberg group