Mathematical Sciences: Partial Differential Equations & Harmonic Analysis

四名调查员将合作开展一个项目， 源于调和理论的数学问题 分析和偏微分方程。 重点将是 上椭圆和抛物边值问题研究 具有相对粗糙边界的域（Lipschitz域）， 关于非光滑椭圆型方程的研究 系数其他工作将集中在变分问题 涉及自由边界，几何问题， 自由薛定谔的曲率和薛定谔乘子 方程 椭圆问题的最新进展导致了尖锐的 椭圆方程组解的估计 流体静力学（斯托克斯系统）和拉梅系统的 弹性静力学，定义在Lipschitz域上。 估计数 定义在整环上的极大函数的相关二次范数 在边界上的解的范数相同。 这项研究的直接目标是， 可能的话，估计到其他的权力规范。 困难一 最基本的德乔治-纳什-莫泽理论 二阶发散形式方程不结转到 系统.此外，实例表明， 大于三个将特别难以实现。 自由边界问题的工作是由以下问题激发的： 液晶理论：液滴，移动缺陷， 向错、相变和流动现象。 的 液晶和其它液晶之间的边界的描述 材料、液体中的线缺陷等特性 需要分析复杂的，奇异的 微分方程组 一种新的模式有待研究 在这项工作中，它承诺简化一些障碍， 在目前的模型中。 多变量非线性椭圆型方程的研究 还将处理几何问题。 这些跌入凡尘 Matukuma方程，标量曲率 方程和非线性平均曲率方程。 第一 与球状星星动力学的爱丁顿模型有关 集群-一个重新引起人们兴趣的问题。 这项工作的一个主要目标是确定 在溶液中可以期待的对称性。 第二个问题是， 在流形上确定度量的问题， 给定标量曲率，而第三个涉及方程 规定平均曲率。 后一个问题涉及 问题解的存在性和正则性问题 毛细作用理论

Four investigators will collaborate on a project directed at mathematical problems originating in the theories of harmonic analysis and partial differential equations. Emphasis will be on the study of elliptic and parabolic boundary value problems on domains with relatively coarse boundaries (Lipschitz domains), and on the study of elliptic equations with non-smooth coefficients. Other work will focus on variational problems involving free boundaries, geometric problems related to curvature and Schrodinger multipliers for the free Schrodinger equation. Recent advances in elliptic problems have led to sharp estimates on solutions of elliptic systems arising in hydrostatics (the Stokes system) and in the Lame systems of elastostatics, defined on Lipschitz domains. The estimates relate quadratic norms of maximal functions defined on the domain with the same norm of the solution restricted to the boundary. The immediate goal of this research is to extend, as far as possible, the estimates to other power norms. The difficulty one encounters is that the fundamental DeGiorgi-Nash-Moser theory for second order divergence form equations does not carry over to systems. Moreover, examples suggest that progress in dimensions greater than three will be especially difficult to achieve. Work on free boundary problems is motivated by questions in the theory of liquid crystals: droplets, moving defects, disclination, phase transition and flow phenomena. The description of the boundaries between liquid crystals and other materials, line defects in the liquid and other characteristics of liquid crystals require the analysis of complex, singular systems of differential equations. A new model is to be studied in this work which promises to simplify several of the obstacles present in current models. Studies of nonlinear elliptic equations related to more geometrical issues will also be undertaken. These fall into three categories, the Matukuma equation, the scalar curvature equations and nonlinear mean curvature equations. The first relates to the Eddington model of the dynamics of globular star clusters - a problem about which there has been renewed interest. One primary goal of this work is to determine the degree of symmetry one can expect in solutions. The second concerns the question of determining metrics on manifolds which produce a given scalar curvature while the third concerns equations prescribing mean curvature. The latter problems involve questions of existence and regularity of solutions to problems in capillarity theory.