Nonlinear elliptic and parabolic problems in analysis and geometry

Columbia University, New York NY

Investigators

Abstract

This project is concerned with the study of nonlinear elliptic and parabolic equations in connection with more complex problems of differential geometry and with physical applications. Such problems include the evolution of a hypersurface in Euclidean (N+1)-space by functions of its principal curvatures, the Ricci flow on both Riemannian and Lorentzian manifolds, the Yamabe flow on surfaces, and the Weyl problem with nonnegative Gaussian curvature. Most of the proposed problems involve equations that are either singular or degenerate, hence problems for which the classical results fail. New analytical techniques will be developed in the project. The project links a wide range of active fields of mathematics, in particular, nonlinear partial differential equations, geometry, and classical analysis. The proposed research activity on the geometry and regularity of degenerate nonlinear parabolic equations and free-boundary problems may lead to significant geometric and even topological applications. The principal investigator intends to study the applications of the mathematical problems to other disciplines such as quantum field theory, relativity theory, plasma physics, image analysis, and thin liquid film dynamics. Results will be disseminated to the research community at various meetings and by publication of research articles. New courses linking partial differential equations and geometric analysis for graduate students will be designed and implemented.