The goal of this work is the development of geometric methods for solving nonlinear partial differential equations (PDE's) arising in problems involving maps with controlled Jacobian determinant. Many problems in differential geometry, optics, and other areas of mathematics and engineering are in this class. Recently discovered deep connections between such equations and Monge-Kantorovich optimal mass transfer theory in Euclidean space and on manifolds will also be studied. Among the topics that will be considered are the following. (1) Development of geometrical and analytic techniques for solving problems requiring determination of reflecting and refracting interfaces with capabilities to transform intensity distributions in a prescribed manner. (2) Investigation of geometric problems involving hypersurfaces with prescribed curvature functions and geometric inequalities with emphasis on variational methods, especially, those connected with Monge-Kantorovich theory; applications of these variational methods to problems in convexity, in particular, to the Minkowski problem and its various generalizations, will be studied as a part of this program. (3) Development of geometrically motivated, provably convergent and efficient multi-scale numerical methods for solving nonlinear second order PDE's arising in reflector/refractor problems of optics and in geometric problems involving curvature functions and maps with controlled volume.
Nonlinear partial differentials equations expressing energy conservation laws as a constraint on the Jacobian of a map describing a physical phenomenon are very common in science and engineering. For example, in optics such equations arise naturally in problems requiring determination of interfaces with prescribed refractive and/or reflective properties; in astrophysics these equations have to be solved when the shape of targets in the solar system must be determined from indirect and limited set of measurements; in weather prediction models based on quasi- and semi-geostrophic approximations of atmospheric motion such equations describe energy conservation laws; in computer science the same type of equations arise in problems connected with radiosity estimates. Typically, the theoretical analysis and numerical solution of these equations is very difficult because of their highly nonlinear structure. Fortunately, the geometric content common to all these problems provides important insights leading to effective methods for their investigation and numerical solution. Development of such methods is the main goal of this research.
