Nonlinear Elliptic and Parabolic Problems

非线性椭圆和抛物问题。拟议研究摘要Panagiota Daskalopoulos这个项目将研究一些椭圆和抛物问题，出现在几何。这些包括超曲面通过其主曲率的函数的演化， Ricci流、Yamabe流和具有非负高斯曲率的Weyl问题。同时也讨论了在点或界面退化或奇异的非线性椭圆方程和抛物方程的可解性。所提出的问题将使用几何技术，涉及奇异性或退化的方程进行研究。要解决的问题包括弱解的存在性，最佳的正则性，并详细分析奇点的形成。第一部分是关于退化的完全非线性椭圆型方程解的最优正则性及相关的自由边界问题的研究。项目的第二部分将调查是否存在 退化的完全非线性几何流的解的最优正则性，包括高度退化的高斯曲率流和调和平均曲率流。这些问题的解决方案的理解可能有重大的几何，甚至拓扑，应用。在项目的第三部分， 将研究快扩散方程的非负解。特别强调的是几何相关的情况下，里奇流和山部流，在那里的奇异性形成的完整度量的非紧表面和相关问题，如永恒的解决方案的分类将被研究。该项目连接在一系列活跃的数学领域的研究-主要是非线性偏微分方程，几何和经典分析。奇异扩散的模型，这将是在这个项目中研究出现在各种物理应用，如人口动力学，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Ricci流和Yamabe流。这些数学领域的不同观点应该联合收割机来进一步照亮其他领域，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.