Geometric PDE's and Monge-Kantorovich Theory in Problems of Optics and Differential Geometry

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DMS-0405622P.I.: Vladimir Oliker, Emory UniversityTitle: Geometric PDE's and Monge-Kantorovich Theory in Problems ofOptics and Differential GeometryABSTRACTThe goal of this work is the development of geometricmethods for solving nonlinear partial differential equations (PDE's)arising in problems involving maps with controlled Jacobiandeterminant. Many problems in differential geometry, optics,and other areas of mathematics and engineering are inthis class. Recently discovered deep connections between such equationsand Monge-Kantorovich optimal mass transfer theory in Euclidean spaceand 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 intensitydistributions 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 andits 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 physicalphenomenon are very common in science and engineering. For example,in optics such equations arise naturally in problemsrequiring determination of interfaces with prescribedrefractive and/or reflective properties; in astrophysics these equationshave to be solved when the shape of targets in the solarsystem must be determined from indirect and limited set of measurements;in weather prediction models based on quasi- and semi-geostrophicapproximations of atmospheric motion such equations describeenergy conservation laws; in computer science the same type ofequations arise in problems connected with radiosity estimates.Typically, the theoretical analysis and numerical solution ofthese equations is very difficult because of their highly nonlinearstructure. Fortunately, the geometric content common to all these problemsprovides important insights leading to effective methods for theirinvestigation and numerical solution. Development of such methodsis the main goal of this research.