Mathematical Sciences: Presidential Young Investigator Award

由总统青年研究者奖资助的工作将主要集中在数学在弹性和液晶问题中的应用，以及与研究具有边界的流形曲率有关的几何问题。要解决的问题是以各种偏微分方程的形式提出的，这些偏微分方程要么是由物理考虑引起的——最小化函数或变分不等式，要么是由几何概念引起的。后者包括在给定里奇张量的流形上求度规的问题。通过对椭圆方程系统的分析，对该问题进行了局部或全局的研究。在Hodge理论中，该问题简化为求解退化椭圆系统解的边界正则性问题。对于单个方程，这个问题最近已经解决了。弹性（塑性）研究的重点将放在Singonolli问题上，寻求解决障碍问题的向量值函数的最佳正则性。在液晶中，许多物理性质，如线奇点、相变和移动缺陷，可以用一个综合的理论来解释。相应的动力学方程的研究将做相关的工作。在各向同性的情况下，这些问题不仅简化了调和映射的热流分析，而且简化了从闵可夫斯基空间到奇异目标流形的调和映射。

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.