Elliptic Boundary Problems and Evolution Equations in Partial Differential Equations

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In this project the principal investigator will tackle problems in two areas in partial differential equations. The first involves equations for functions that vary with space but not with time. Such stationary problems model many physical systems in equilibrium, from static electric fields to configurations of elastic bodies, and also arise in fundamental investigations in mathematical analysis, including potential theory and analytic function theory. Problems of this type often involve unknown functions defined on bounded regions. One goal of this project is to extend our knowledge of how to handle such problems when the boundaries are quite rough. The second area involves evolution equations, for functions that vary with time. Core classes of interest in this project include equations for wave motion, both for classical waves and variants, such as Schrodinger equations and Dirac equations, whose origins lie in the motions of atoms.This project will develop tools to advance the study of elliptic systems on domains with uniformly rectifiable boundary, which is essentially the maximal class of domains on which one can use singular integral operator techniques. One class of problems that will be tackled consists of Riemann-Hilbert type problems. These were first studied on planar domains, with piecewise smooth interfaces. This project will develop higher dimensional versions of Riemann-Hilbert problems, on domains with uniformly rectifiable interfaces. A related study will involve a development of the index theory of Toeplitz operators on such rough domains. The other major part of this project concerns evolution equations. Specific problems to be tackled include studies of wave decay, via various mechanisms, some related to the formation of harmonics on stringed instruments, but in a higher dimensional context. In addition, wave decay problems will be considered on domains with rough geometry, including cases where the main geometrical hypothesis is a lower bound on the Ricci tensor, and the challenge is to develop a theory of geometrical optics in this setting.