Elliptic Boundary Value Problems in Non-Smooth Domains

非光滑区域上的椭圆型边值问题。研究摘要沈中伟本研究项目主要研究具有非光滑边界的区域上的偏微分方程组问题。PI将研究一类有界Lipschitz区域上边值问题的可解性。这是一类伸缩不变的区域，其边界是Lipschitz函数的图形。主要集中在具有Lp边界数据的二阶椭圆组和高阶椭圆型方程边值问题。他还将研究凸域中的边值问题。这项研究位于调和分析和偏微分方程组的交界处。该项目的目标是在区域和边界数据的物理现实假设下建立解的有用的正则性估计。在弹性力学、空气动力学、流体力学和电磁波散射等许多应用问题中，控制方程的边值问题都是在具有粗糙边界的区域中提出的。该项目的结果将为这些应用提供一些数学基础和分析工具。

Elliptic Boundary value problems in non-smooth domains.Abstract of proposed researchZhongwei ShenThis research project centers on problems in the area of partial differential equations in domains with non-smooth boundaries. The PI will study the solvability of boundary value problems on the class of bounded Lipschitz domains. This is a dilation-invariant class of domains which have boundaries that are the graphs of Lipschitz functions. The main focus will be on boundary value problems for second order elliptic systems and higher-order elliptic equations with Lp boundary data. He will also investigate boundary value problems in convex domains.This research lies at the interface of harmonic analysis and partial differential equations.The goal of the project is to establish useful regularity estimates for solutions under physically realistic assumptions on the domain as well as on the boundary data. In many applied problems of elasticity, aero- and hydrodynamics and electro-magnetic wave scattering, the boundary value problems for the governing equations are posed in domains with rough boundaries. The results of this project will provide some mathematical foundations and analytical tools for these applications.