Mathematical Sciences: Partial Differential Equations & Harmonic Analysis

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Four investigators will collaborate on a project directed at mathematical problems originating in the theories of harmonic analysis and partial differential equations. Emphasis will be on the study of elliptic and parabolic boundary value problems on domains with relatively coarse boundaries (Lipschitz domains), and on the study of elliptic equations with non-smooth coefficients. Other work will focus on variational problems involving free boundaries, geometric problems related to curvature and Schrodinger multipliers for the free Schrodinger equation. Recent advances in elliptic problems have led to sharp estimates on solutions of elliptic systems arising in hydrostatics (the Stokes system) and in the Lame systems of elastostatics, defined on Lipschitz domains. The estimates relate quadratic norms of maximal functions defined on the domain with the same norm of the solution restricted to the boundary. The immediate goal of this research is to extend, as far as possible, the estimates to other power norms. The difficulty one encounters is that the fundamental DeGiorgi-Nash-Moser theory for second order divergence form equations does not carry over to systems. Moreover, examples suggest that progress in dimensions greater than three will be especially difficult to achieve. Work on free boundary problems is motivated by questions in the theory of liquid crystals: droplets, moving defects, disclination, phase transition and flow phenomena. The description of the boundaries between liquid crystals and other materials, line defects in the liquid and other characteristics of liquid crystals require the analysis of complex, singular systems of differential equations. A new model is to be studied in this work which promises to simplify several of the obstacles present in current models. Studies of nonlinear elliptic equations related to more geometrical issues will also be undertaken. These fall into three categories, the Matukuma equation, the scalar curvature equations and nonlinear mean curvature equations. The first relates to the Eddington model of the dynamics of globular star clusters - a problem about which there has been renewed interest. One primary goal of this work is to determine the degree of symmetry one can expect in solutions. The second concerns the question of determining metrics on manifolds which produce a given scalar curvature while the third concerns equations prescribing mean curvature. The latter problems involve questions of existence and regularity of solutions to problems in capillarity theory.