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

本项目所要分析的问题来源是 从微分几何和数学物理学。 这些领域 提供了非线性方程的重要原型 其中必须进行基本和困难分析， 了解溶液的性质。 工作将集中在 奇异性和正则性问题的研究 偏微分方程的解，例如 在几何学和理论中的Yamabe方程中遇到的 的液晶。 其他工作将考虑边界 非常退化的非线性椭圆方程组的正则性， 出现在努力决定什么时候可以找到谐波 两个域（不一定平滑）之间的映射， 规定了边界值。 这项工作将侧重于几个具体目标。 其中 将研究非线性椭圆方程的奇异解 方程的努力，以获得估计的尺寸 奇异集以及理解这样的位置 集. 在奇异集是孤立的情况下，它的位置 似乎是僵硬的。 在液晶模型的研究中， 将完成检查之间的过渡， 近晶态的晶体配置，而更多的几何 工作将寻求提供信息，如何才能确定 紧黎曼流形是否共形 相当于一个球体。

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.