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

本项目的重点是不规则的几何形状和内部介质的不均匀性的基本边值问题的解决方案的行为的影响。尽管两个多世纪的深入研究和大量的结果，今天的偏微分方程的域被彻底理解只有当系数和域表现出光滑性。 然而，在真实的生活中的主导地位的不规则性和现代数学中的许多问题的经典“光滑”理论的扩展的要求。PI所追求的三个主要方向是：（a）非光滑域上粗糙数据边界问题的研究;（B）非光滑系数的二阶椭圆算子;（c）高阶椭圆偏微分方程。拟议的工作范围包括全方位的尖锐的适定性和正则性结果，渐近公式，新的措施，容量和新的功能空间，数值实验，本地化现象的介绍和分析。鉴于无处不在的不规则性在自然界和人造机制，他们的理解是必不可少的科学，工程，工业和社会的一般。事实上，大多数经典模型只提供了一个很好的，平滑的近似对象和过程。然而，完全一致的光滑系统在自然界中并不存在，每个真实的物体都无意中拥有奇点（边界的尖锐边缘，介质的突变，结构的缺陷）。它们通常对发生的现象具有决定性影响，但无法在平稳的设置中捕获（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，旨在推进非光滑介质中偏微分方程的分析，并为相关领域的数学家做好准备。后一个目标将特别通过一个新的妇女参与分析和方案主任讲习班来实现。

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.