Nonlinear Diffusion Equations and Free-Boundary Problems

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Title: Nonlinear Diffusion Equations and Free boundary problemsPI: Panagiota Daskalopoulos, Columbia UniversityABSTRACTThis 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 WeylProblem 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 ofquasilinear and fully-nonlinear geometric flows,such as the Gauss curvature flow withflat sides or more general curvature flows including the Harmonic flow.The understanding of such models of equations and free-boundary problemsmay have significant geometric and even topological applications.A different new line of research will study the regularity of solutionsofdegenerate Monge-Ampere equations and related ellipticfree-boundary problems. Its main goal is to develop new techniques toestablish the optimal regularity in fully-nonlineardegenerate elliptic equations. The proposed work is also motivated bythe well known Weyl problem with nonnegative Gaussian curvature.The aim of the third part of the project is to study the connectionbetween the geometry and the regularity as well as the formation ofsingularities in Stefan type free-boundary problems including also theHele-Shaw flow and free-boundary problem in flame propagation. The useof the geometric aspects of the problems is crucial in the proposedapproach. The last part of the proposed activity will study the asymptoticbehavior 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 conformalmetrics on a non-compact surface.This project links a wide range of active fields of mathematics, inparticular nonlinear partial differential equations, geometry andclassical analysis.The proposed research activity on the geometry and regularity ofdegenerate nonlinear parabolic equations and free-boundary problems mayresult to significant geometric and even topological applications. Theproposed research activity on Stefan type free-boundary problems isclosely relatedto various important physical models, including the propagation of thepremixed equi-diffusional flames in the limit of high activation energy.The models of singular diffusion which will be studiedin this project arise in variousphysical applications such as population dynamics, the kinetic theory ofgases and thin liquid film dynamics.Students and postdocs will be trained as part of this project. Special emphasis will be given to theencouragement of talented femaleundergraduate students, graduate students and postdocsto pursue a successfulcareer in mathematics or related sciences. New courses linking PartialDifferential Equations and Geometric Analysis for graduate students will be designedand implemented.