Elliptic Boundary Problems and Evolution Equations in Partial Differential Equations

在这个项目中，首席研究员将解决偏微分方程式的两个领域的问题。第一个涉及随空间变化但不随时间变化的函数的方程。这种静态问题模拟了许多处于平衡状态的物理系统，从静电场到弹性体的构型，也出现在数学分析的基础研究中，包括位势理论和解析函数理论。这类问题通常涉及定义在有界区域上的未知函数。这个项目的一个目标是扩展我们在边界相当粗糙的情况下如何处理这些问题的知识。第二个领域涉及随时间变化的函数的演化方程。这个项目的核心类包括波动方程，包括经典波和变种的波动方程，例如薛定谔方程和狄拉克方程，它们的起源在于原子的运动。这个项目将开发工具来推进具有一致可校正边界的区域上的椭圆系统的研究，这本质上是可以在其上使用奇异积分算子技术的最大类区域。要解决的一类问题是Riemann-Hilbert型问题。这些研究首先是在具有分段光滑界面的平面区域上进行的。这个项目将在具有统一可校正界面的区域上开发Riemann-Hilbert问题的高维版本。相关的研究将涉及Toeplitz算子在这种粗糙区域上的指标理论的发展。这个项目的另一个主要部分是关于进化方程。需要解决的具体问题包括通过各种机制研究波的衰减，其中一些与弦乐器上的谐波形成有关，但在更高维度的背景下。此外，波的衰减问题将被考虑在具有粗糙几何的区域上，包括主要几何假设是Ricci张量的下界的情况，而挑战是在这种情况下发展几何光学理论。

In this project the principal investigator will tackle problems in two areas in partial differential equations. The first involves equations for functions that vary with space but not with time. Such stationary problems model many physical systems in equilibrium, from static electric fields to configurations of elastic bodies, and also arise in fundamental investigations in mathematical analysis, including potential theory and analytic function theory. Problems of this type often involve unknown functions defined on bounded regions. One goal of this project is to extend our knowledge of how to handle such problems when the boundaries are quite rough. The second area involves evolution equations, for functions that vary with time. Core classes of interest in this project include equations for wave motion, both for classical waves and variants, such as Schrodinger equations and Dirac equations, whose origins lie in the motions of atoms.This project will develop tools to advance the study of elliptic systems on domains with uniformly rectifiable boundary, which is essentially the maximal class of domains on which one can use singular integral operator techniques. One class of problems that will be tackled consists of Riemann-Hilbert type problems. These were first studied on planar domains, with piecewise smooth interfaces. This project will develop higher dimensional versions of Riemann-Hilbert problems, on domains with uniformly rectifiable interfaces. A related study will involve a development of the index theory of Toeplitz operators on such rough domains. The other major part of this project concerns evolution equations. Specific problems to be tackled include studies of wave decay, via various mechanisms, some related to the formation of harmonics on stringed instruments, but in a higher dimensional context. In addition, wave decay problems will be considered on domains with rough geometry, including cases where the main geometrical hypothesis is a lower bound on the Ricci tensor, and the challenge is to develop a theory of geometrical optics in this setting.