Mathematical Sciences: Presidential Young Investigator Award

这项由总统青年研究员奖赞助的工作将主要集中在数学应用于弹性问题和液晶问题，以及与研究有边界流形上的曲率有关的几何问题上。所要解决的问题以各种偏微分方程式的形式提出，这些偏微分方程系由物理上的考虑--泛函的最小化或变分不等，或由几何概念所引起。后者包括在具有规定的Ricci张量的流形上寻找度量的问题。这个问题是通过对椭圆型方程组的分析来局部或全局研究的。在Hodge理论中，问题归结为求解退化椭圆组的解的边界正则性问题。对于单一的方程，这个问题最近已经解决了。弹性(塑性)研究的焦点将集中在Singonolli问题上，寻求解决障碍问题的向量值函数的最佳可能正则性。在液晶中，许多物理性质，如线奇点、相变和运动缺陷，都可以用一个全面的理论来解释。相关工作将在相应的动力学方程的研究上进行。在各向同性的情况下，这些问题归结为对调和映射的热流分析以及从Minkowski空间到奇异目标流形的调和映射的分析。

Work sponsored by this Presidential Young Investigator Award will focus primarily on questions arising in the applications of mathematics to problems of elasticity and liquid crystals, to geometric problems related to the study of curvature on manifolds with boundary. The problems to be addressed are presented in terms of various partial differential equations arising either from physical considerations - minimization of functionals or variational inequalities, or are motivated by geometrical concepts. The latter include the problem of finding a metric on a manifold with prescribed Ricci tensor. This problem is studied locally or globally through the analysis of systems of elliptic equations. In the Hodge theory, the problem reduces to that of solving the boundary regularity of solutions of degenerate elliptic systems. For a single equation, the problem has recently been settled. Focus of the elasticity (plasticity) studies will be on the Singonolli problem, seeking the best possible regularity of vector valued functions which solve obstacle problems. In liquid crystals many physical properties such as line singularities, phase transition, and moving defects, may be explained within a comprehensive theory. Related work will be done on studies of the corresponding dynamical equations. In an isotropic case, these problems reduce tot he analysis of heat flow of harmonic maps and to harmonic maps from Minkowski space to singular target manifolds.