Topics in the Theory of Elliptic Boundary Value Problems

Intellectual Merit. The intellectual focus of the present proposal is in the area of intersection between harmonic analysis and partial dierential 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 (domains with either isolated singularities or Lipschitz domains). While elliptic boundary value problems for the Laplacian have been well-understood for over 20 years, the theory for second order elliptic systems and higher order elliptic equations is not fully developed. The three main research directions of the proposal are brie y described below. 1. Boundary value problems for higher order elliptic operators. The layer potential method has proved to be very successful in the treatment of second order elliptic boundary value problems, but currently this approach is insuciently developed to deal with the intricacies of the theory of higher order operators. Building on recent progress in the employment of potential methods for the Neumann problem for the biharmonic equation on non-smooth domains, this project will undertake a systematic treatment of the spectral properties of the layer potential operators naturally associated with general higher order elliptic boundary value problems in smooth and non-smooth domains in two dimensions. 2. The spectral radius conjecture on Besov spaces. Solving boundary value problems via the method of layer potentials typically reduces the original problem to that of inverting an operator of the form I + K on appropriate (boundary) function spaces, where I is the identity operator. This research theme examines conditions under which the spectrum of K on acting on Besov spaces is strictly less than 1. Here, the operator K is the boundary version of the double layer potential operator associated to the Laplacian or the Lame system. For arbitrary Lipschitz domains, this issue is occasionally referred to as the spectral radius conjecture. 3. Well-posedness of the mixed problem. The Dirichlet and Neumann boundary value problems are special cases of the more general situation when mixed boundary conditions are considered. This research theme investigates open problems related to the well-posedness of the mixed problem for the Laplacian and the Lame system of elastostatics. Specically, the investigation will focus on the following two topics: a) sharp well-posedness of the mixed problem in Lipschitz domains; b) the mixed problem for the Lame system of elastostatics. Broader Impacts. The research themes of this proposal are motivated by problems that naturally arise in mathematical physics and engineering. In this regard, the non-smooth setting in which these problems are posed is fundamental since most realistic physical models involve irregular domains. For example, (A) has applications to engineering in the context of modeling shallow shells, beam bending, and clamped plates. Problems falling under (C) model the behavior of several physical quantities such as the temperature in a metallurgical melting process, the thermoelastic potential of an elastic solid punched or stamped by a heated object, or the seepage through a porous material. In parallel with pursuing the research directions outlined above, the proposer intends to con- tinue her commitment to activities aimed at increasing the participation of women and other under-represented groups in mathematics. Two examples of these outreach activities are the Sonia Kovalevsky High School Mathematics Day and the AWM presence at the USA Science and Engi- neering Festival Expo. In the classroom, the PI will continue to develop a proof-writing course in Number Theory for sophomores and juniors majoring in mathematics to help ease the transition to upper division mathematics courses. The proposer will also seek to popularize and promote mathematics by publishing expository articles related to mathematics. Recently, her articles have appeared in the Notices of the AMS, the AWM Newsletter, the magazine Science, and SIAM News. 1