The proposed project is aimed at a wide range of problems in the theory of elliptic partial differential equations on rough domains. The ultimate goal is to understand the intricate relations between the geometry of the domain, the nature of the data, the structure of the equation, and the regularity of the solutions. Among other problems, the project addresses fundamental properties of solutions to the higher order elliptic equations in arbitrary domains (such as maximum principle and Wiener criterion), sharp estimates on the solutions of Dirichlet and Neumann problems in terms of the data in the presence of boundary singularities, as well as elliptic operators with complex bounded measurable coefficients. The research plan incorporates techniques originating from different branches of modern analysis (harmonic analysis, operator and spectral theory, function spaces) and promotes the development of new methods, which unravel some completely new phenomena.
The elliptic problems naturally arise in various branches of physics, such as electrostatics, thermodynamics, and elasticity. However, despite its long history, the theory of elliptic partial differential equations contains many open questions. This work will contribute to further progress in the aforementioned areas of science and engineering, and the results of the proposed research will be disseminated at various levels: through publications and presentations in national and international professional meetings, communication with researchers in analysis and other fields, formal and informal educational and outreach activities.
