Mathematical Sciences: Partial Differential Equations & Harmonic Analysis

四名研究人员将合作一个项目，针对起源于调和分析和偏微分方程式理论的数学问题。重点是研究具有相对粗糙边界的区域(Lipschitz域)上的椭圆和抛物型边值问题，以及研究具有非光滑系数的椭圆型方程。其他工作将集中在涉及自由边界的变分问题，与曲率有关的几何问题，以及自由薛定谔方程的薛定谔乘子。椭圆问题的最新进展导致了在流体静力学(Stokes系统)和定义在Lipschitz域上的弹性静力学的Lame系统中产生的椭圆组的解的精确估计。这些估计将定义在区域上的极大函数的二次范数与受限于边界的解的相同范数联系起来。这项研究的直接目标是尽可能地将估计扩展到其他电力规范。我们遇到的困难是，二阶散度形式方程的基本DeGiorgi-Nash-Moser理论不能推广到系统。此外，例子表明，在三个维度以上的进展将特别难以实现。关于自由边界问题的工作是由液晶理论中的问题推动的：液滴、运动缺陷、错位、相变和流动现象。描述液晶和其他材料之间的边界、液体中的线缺陷和液晶的其他特性需要分析复杂的、奇异的微分方程组。这项工作将研究一种新的模型，它有望简化目前模型中存在的几个障碍。还将研究与更多几何问题有关的非线性椭圆型方程。它们分为三类：Matukuma方程、标量曲率方程和非线性平均曲率方程。第一个与球状星团动力学的爱丁顿模型有关--这个问题重新引起了人们的兴趣。这项工作的一个主要目标是确定人们在解决方案中可以预期的对称度。第二个涉及产生给定标量曲率的流形上的度量的确定问题，而第三个涉及规定平均曲率的方程。后一类问题涉及毛细理论中问题解的存在性和正则性问题。

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.