Analytical and Geometrical Problems in Non Linear Partial Differential Equations

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Analytical and Geometrical Problems in Nonlinear Partial Differential EquationsAbstract of Proposed Research Luis CaffarelliThis research is to study the mathematical analysis of a number of scientific phenomena that are modeled by nonlinear partial differential equations. Specific topics include the properties of solutions of nonlinear problems involving anomalous (in particular integral) diffusion processes, such as phase transitions, fluid dynamics and optimal control. Also nonlinear random homogenization of fully non linear equations or constrained problems in randomly perforated domains. Antenna design for general signal propagation laws, and other Minkowski type problemsSpecific examples of the type of phenomena being studied include the analysis of equations modeling boundary control (optimizing insulation shape across a surface, or the behavior of semi permeable membranes), surface flame propagation and the pricing of options when processes are highly discontinuous ( Levi processes). Also the effective heating of a family of small, randomly distributed, heating sources, the propagation of a flame on a medium with random occlusions or the sliding of a drop on a rough surface and the design of an optimal reflecting surface in a periodic medium.