Nonlinear Diffusion Equations and Free-Boundary Problems

Columbia University, New York NY

Investigators

Abstract

Title: Nonlinear Diffusion Equations and Free boundary problems PI: Panagiota Daskalopoulos, Columbia University ABSTRACT This project concerns with the study of nonlinear elliptic and parabolic equations and free-boundary problems, in connection with more complex problems of differential geometry,includingthe Gauss curvature flow, the Ricci flow and the Weyl Problem withnonnegative Gaussian curvature and with physical applications such as thin liquidfilm dynamics and flame propagation.The first part of the project will study thegeometry and regularity of free-boundary problems arising from the degeneracy of quasilinear and fully-nonlinear geometric flows,such as the Gauss curvature flow with flat sides or more general curvature flows including the Harmonic flow. The understanding of such models of equations and free-boundary problems may have significant geometric and even topological applications. A different new line of research will study the regularity of solutions ofdegenerate Monge-Ampere equations and related elliptic free-boundary problems. Its main goal is to develop new techniques to establish the optimal regularity in fully-nonlinear degenerate elliptic equations. The proposed work is also motivated by the well known Weyl problem with nonnegative Gaussian curvature. The aim of the third part of the project is to study the connection between the geometry and the regularity as well as the formation of singularities in Stefan type free-boundary problems including also the Hele-Shaw flow and free-boundary problem in flame propagation. The use of the geometric aspects of the problems is crucial in the proposed approach. The last part of the proposed activity will study the asymptotic behavior of solutions of variousmodels of singular diffusion. In particular, it will deal with the type II blow up behavior of maximal solutions of the two dimensionalRicci Flow. These solutions correspond to complete Riemannian conformal metrics on a non-compact surface. This 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 result to significant geometric and even topological applications. The proposed research activity on Stefan type free-boundary problems is closely relatedto various important physical models, including the propagation of the premixed equi-diffusional flames in the limit of high activation energy. 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. Students and postdocs will be trained as part of this project. Special emphasis will be given to the encouragement of talented femaleundergraduate students, graduate students and postdocs to pursue a successfulcareer in mathematics or related sciences. New courses linking PartialDifferential Equations and Geometric Analysis for graduate students will be designed and implemented.