CAREER: Analysis of Partial Differential Equations in non-smooth media

本项目的重点是不规则几何和内部介质不均匀对基本边值问题解的行为的影响。尽管经过了两个多世纪的深入研究和大量的结果，今天只有当系数和区域都表现出光滑性时，区域上的偏微分方程才能被彻底理解。然而，现实生活中的不规则性和现代数学中的大量问题要求对经典的“光滑”理论进行扩展。PI所追求的这一扩展的三个主要方向是：(A)研究非光滑区域上具有粗糙数据的边界问题；(B)具有非光滑系数的二阶椭圆算子；(C)高阶椭圆型偏微分方程组。拟议工作的范围包括全面的适定性和正则性结果、渐近公式、新度量、新容量和新函数空间的引入和分析、数值实验、局部化现象。鉴于自然界和人为机制中的不规则性普遍存在，对它们的理解对科学、工程、工业和整个社会都是至关重要的。事实上，大多数经典模型只提供对象和过程的良好、平滑的近似值。然而，自然界中并不存在完全均匀的光滑系统，每个真实的物体都无意中具有奇点(边界的尖锐边缘、介质的突变、结构的缺陷)。它们往往对发生的现象具有决定性的影响，但不能在平滑的设置中捕捉到(例如，肺的分形结构)。本项目结合了数学追求和严格的教育计划，朝着推进偏微分方程在非光滑介质中的分析和培养在相关领域工作的不同数学家的总目标。后一项目标尤其将通过一个新颖的妇女分析和PDE讲习班来实现。

The present project is focused on the impact of irregular geometry and internal medium inhomogeneity on the behavior of solutions to fundamental boundary value problems. Despite over two centuries of intensive research and a massive body of results, today partial differential equations on domains are thoroughly understood only when both the coefficients and the domain exhibit smoothness. However, the dominance of irregularities in real life and numerous problems in modern mathematics demand for an extension of the classical "smooth" theory. Three major directions for this extension pursued by the PI are: (a) the study of boundary problems with rough data on non-smooth domains; (b) the second-order elliptic operators with non-smooth coefficients; (c) elliptic PDEs of higher order. The scope of the proposed work incorporates a full range of sharp well-posedness and regularity results, asymptotic formulas, introduction and analysis of new measures, capacities and new function spaces, numerical experiment, localization phenomena.Given the ubiquitousness of irregularities in nature and in man-made mechanisms, their understanding is essential to science, engineering, industry, and to society in general. Indeed, most classical models only provide a nice, smooth approximation of the objects and processes. However, perfectly uniform smooth systems do not exist in nature, and every real object inadvertently possesses singularities (a sharp edge of the boundary, an abrupt change of the medium, a defect of the construction). They often have a decisive effect on the occurring phenomena, but cannot be captured in a smooth set-up (e.g., the fractal structure of lungs). The present project combines the mathematical pursuit and a rigorous educational program towards the general goal of advancing analysis of partial differential equation in non-smooth media and preparing a diverse array of mathematicians working in related areas. The latter objective will, in particular, be achieved through a novel Workshop for Women in Analysis and PDE.