Ellilptic Boundary Value Problems in Non-Smooth Domains

The research described in this proposal centers on problems in the area of partial differential equations in domains with non-smooth boundaries. In many applied problems of elasticity, aero- and hydrodynamics, and electro-magnetic wave scattering, the boundary value problems for the partial differential equations are posed in domains with rough boundaries. The PI will study the solvability of the boundary value problems on the class of bounded Lipschitz domains. This is a dilation-invariant class of domains whose boundaries are locally given by the graphs of Lipschitz functions. The main focus will be on the second order elliptic systems and higher-order elliptic equations. Although the boundary value problems with L2 boundary data were solved about twenty years ago, the Lp boundary value problems for elliptic systems and higher-order elliptic equations remains open in dimension four or higher. The PI will also investigate the boundary value problems in convex domains. The goal of the project is to establish useful regularity estimates of solutions under physically realistic assumptions on the domain as well as on the boundary data. The results of this project will provide mathematical foundation and analytical tools for certain engineering applications. The proposed research lies at the interface of harmonic analysis and and partial differential equations. The findings from the research will be disseminated in the scientific community through presentations in conferences and publishing in mathematical journals and websites. Funds are requested to support graduate students as research assistants. The proposed research activities will take place in the state of Kentucky, which is covered by the EPSCoR (Experimental Program to Stimulate Competitive Research). The development of research infrastructure and enhancing science research capabilities in Kentucky are consistent with the goals of the EPSCoR project