Elliptic Boundary Value Problems in Non-Smooth Domains

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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.