Nonlinear Elliptic and Parabolic Problems

非线性椭圆型和抛物型问题。拟研究摘要Panagiota daskalopoulos这个项目将研究几何学中出现的一些椭圆型和抛物型问题。这包括一个由其主曲率函数的超曲面的演化，Ricci流，Yamabe流，和非负高斯曲率的Weyl问题。也讨论了在点或界面处退化或奇异的非线性椭圆型和抛物型方程的可解性。所提出的问题将使用涉及方程的奇异性或简并性的几何技术来研究。要解决的问题包括弱解的存在性、最优正则性以及对奇点形成的详细分析。本文第一部分研究了退化的全非线性椭圆型方程解的最优正则性及其相关的自由边界问题。项目的第二部分将研究退化的完全非线性几何流的解的存在性和最优正则性，包括高度退化的高斯曲率流和调和平均曲率流。对这些问题的解的理解可能具有重要的几何甚至拓扑应用。在项目的第三部分，将研究快速扩散方程的非负解的消光行为。特别强调在几何上相关的Ricci流和Yamabe流的情况下，在非紧曲面上完全度量的奇点形成和相关的问题，如永恒解的分类将被研究。该项目将一系列活跃的数学领域的研究联系起来-主要是非线性偏微分方程，几何和经典分析。本项目将研究的奇异扩散模型出现在各种物理应用中，如种群动力学、气体动力学理论和薄液膜动力学。它们也出现在微分几何中，如曲面上的里奇流和山部流。这些数学领域的不同观点应该结合起来进一步阐明其他领域，并帮助解决重要的几何问题。

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.