Topics in the Theory of Elliptic Boundary Value Problems

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This mathematics research project focuses on problems in harmonic analysis and partial differential equations, with an emphasis on the study of elliptic boundary value problems for second and higher order operators and for systems in non-smooth domains. A non-smooth domain refers to a domain with either isolated singularities or a Lipschitz domain. While elliptic boundary value problems for the Laplacian have been well understood for over twenty years, the theory for second order elliptic systems and higher order elliptic equations is incomplete. Another generalization of the classical Dirichlet and Neumann boundary value problems that is not yet fully understood is the mixed boundary value problem. This project addresses open problems that are categorized by the following research themes: boundary value problems for higher order elliptic operators; the spectral radius conjecture on Besov spaces; well-posedness of the mixed problem in Lipschitz domains. This mathematics research project is motivated by problems that naturally arise in mathematical physics and engineering. In this regard, the non-smooth setting in which the problems are posed is fundamental since most realistic physical models involve irregular domains. For example, boundary value problems for higher order elliptic operators have applications to engineering in the context of modeling shallow shells, beam bending, and clamped plates. Mixed boundary value problems model the behavior of several physical quantities such as the temperature in a metallurgical melting process, the thermo-elastic potential of an elastic solid punched or stamped by a heated object, or the seepage through a porous material. This mathematics research project will contribute to the aforementioned disciplines as well as produce new mathematical techniques. While pursuing the research directions outlined above, Ott will initiate activities aimed at increasing the participation of women and other under-represented groups in mathematics. These activities will include outreach activities with local middle and high schools, and research and networking opportunities for undergraduate and graduate students.