Mathematical Sciences: Singularity and Regularity of Solutions of Scalar Equations and Systems in Connection with Differential Geometry

在这个项目中要分析的问题的来源来源于微分几何和数学物理。这些科学领域提供了非线性方程的重要原型，其中必须进行基本和困难的分析才能理解解的性质。工作将集中于研究与偏微分方程解的奇异性和规律性有关的问题，例如几何中的Yamabe方程和液晶理论中遇到的问题。其他工作将考虑非常退化的非线性椭圆系统的边界规则性，当边界值被规定时，这些系统在决定何时可能找到两个（不一定是光滑的）域之间的调和映射时出现。这项工作将集中在几个具体目标上。其中将包括研究非线性椭圆方程的奇异解，以努力获得奇异集的维数估计以及了解这些集的位置。在奇异集是孤立的情况下，它的位置似乎是刚性的。在液晶模型的研究中，将研究晶体构型的向列态和拟聚态之间的转换，而更多的几何工作将寻求提供有关如何确定紧致黎曼流形是否与球体共形等效的信息。

The source of problems to be analyzed in this project derive from differential geometry and mathematical physics. These areas of science provide important prototypes of non-linear equations in which fundamental and difficult analysis must be done to understand the properties of solutions. Work will concentrate on the study of problems related to singularity and regularity of solutions of partial differential equations such as those encountered in the Yamabe equation in geometry and in the theory of liquid crystals. Other work will consider the boundary regularity of very degenerate non-linear elliptic systems which arise in efforts to decide when it is possible to find harmonic maps between two (non necessarily smooth) domains when the boundary values are prescribed. The work will focus on several specific goals. Among them will be the study of singular solutions of nonlinear elliptic equations in an effort to obtain estimates on the dimension of the singular set as well as to understand the location of such sets. In cases where the singular set is isolated, its location appears to be rigid. In studies of liquid crystal models, work will be done examining the transition between the nematic and smectic states of crystal configurations, while more geometric work will seek to provide information as to how one can determine whether or not a compact Riemannian manifold is conformally equivalent to a sphere.