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。这类问题包括欧氏（N+1）空间中一个超曲面的主曲率函数演化、黎曼流形和洛伦兹流形上的Ricci流、曲面上的Yamabe流以及非负高斯曲率的Weyl问题。大多数提出的问题都涉及到奇异或退化的方程，因此经典结果无法解决的问题。该项目将开发新的分析技术。该项目涉及广泛的数学活跃领域，特别是非线性偏微分方程、几何和经典分析。提出的关于退化非线性抛物方程和自由边界问题的几何和正则性的研究活动可能会导致重要的几何甚至拓扑应用。主要研究数学问题在量子场论、相对论、等离子体物理、图像分析、液体薄膜动力学等学科中的应用。研究结果将通过各种会议和研究论文的出版向研究界传播。为研究生设计并实施将偏微分方程与几何分析联系起来的新课程。

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.