Elliptic Boundary Value Problems in Non-Smooth Domains

非光滑域上的椭圆边值问题。申忠伟本课题主要研究非光滑边界域上的偏微分方程问题。PI将研究一类有界Lipschitz域上边值问题的可解性。这是一类膨胀不变的区域它的边界是利普希茨函数的图。主要的焦点将是二阶椭圆系统的边值问题和具有Lp边界数据的高阶椭圆方程。他还将研究凸域的边值问题。本研究是调和分析与偏微分方程的结合。该项目的目标是在域和边界数据的物理现实假设下，为解决方案建立有用的规则估计。在弹性力学、气动和流体力学以及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。该项目的研究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.