Nonlinear elliptic and parabolic problems in analysis and geometry

这个项目是关于非线性椭圆和抛物方程的研究，与更复杂的微分几何问题和物理应用有关。这些问题包括欧氏（N+1）空间中超曲面通过其主曲率的函数的演化、黎曼流形和洛伦兹流形上的Ricci流、曲面上的Yamabe流以及具有非负高斯曲率的Weyl问题。大多数提出的问题涉及的方程是奇异或退化，因此经典的结果失败的问题。该项目将开发新的分析技术。该项目将广泛的数学活跃领域联系起来，特别是非线性偏微分方程，几何和经典分析。退化的非线性抛物方程和自由边界问题的几何和正则性的拟议的研究活动可能会导致显着的几何，甚至拓扑应用。主要研究者打算研究数学问题在其他学科中的应用，如量子场论，相对论，等离子体物理，图像分析和薄液膜动力学。研究结果将在各种会议上和通过发表研究文章向研究界传播。将为研究生设计和实施连接偏微分方程和几何分析的新课程。

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.