Nonlinear Elliptic and Parabolic Problems

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Nonlinear Elliptic and Parabolic Problems.Abstract of Proposed Research Panagiota DaskalopoulosThis project will study a number of elliptic and parabolic problems that arise in geometry. These include the evolution of a hyper-surface by functions of its principal curvatures, the Ricci flow, the Yamabe flow, and the Weyl Problem with nonnegative Gaussian curvature. Also the solvability of nonlinear elliptic and parabolic equations that either are degenerate, or singular, at points or interfaces. The proposed problems will be studied using geometric techniques that take involve the singularity or degeneracy of the equations. Questions to be addressed include the existence of weak solutions, the optimal regularity, and a detailed analysis of the formation of singularities. The fisrt part of the proposal concerns with the optimal regularity of solutions of degenerate fully-nonlinear elliptic equations and the study of related free-boundary problems. The second part of the project will investigate the existence and optimal regularity of solutions of degenerate fully nonlinear geometric flows, including the highly degenerate Gauss curvature flow and Harmonic mean curvature flows. The understanding of the solutions of these problems may have significant geometric, and even topological, applications. In the third part of the project, the extinction behavior of non-negative solutions of fast-diffusion equations will be investigated. Special emphasis is given to the geometrically relevant cases of the Ricci flow and the Yamabe flow, where the singularity formation of complete metrics on non-compact surfaces and related problems such as the classification of eternal solutions will be studied.This project links research in a range of active mathematical fields - primarily nonlinear partial differential equations, geometry and classical analysis. The models of singular diffusion which will be studied in this project arise in various physical applications such as population dynamics, the kinetic theory of gases and thin liquid film dynamics. They also arise in differential geometry as the Ricci flow and the Yamabe flow on surfaces. The different perspectives of each of these mathematical fields should combine to further illuminate other areas and help solve important geometrical questions.