Mathematical Sciences: Singular Integrals and Parabolic Partial Differential Equations

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9400782 Hofmann This award supports mathematical research on problems in singular integral theory and partial differential equations which arise in connection with boundary value problems for parabolic equations on non-smooth domains, whose boundaries are allowed to vary with time. The problems are motivated in part by physical considerations, such as the expansion and contraction or otherwise changing of shape of materials over time, and in part by mathematical considerations. The time-varying domains which one would like to ultimately treat are in a natural sense theparabolic analogues of the Lipschitz domains which have played a prominent role in the elliptic theory in recent years. The main goal of the project is to solve initial boundary values problems in this context by means of layer potentials. Among the tools which are likely to be employed or studied are the Rellich identities, perturbation techniques, caloric measure estimates, analysis of multilinear and nonlinear singular integrals and the circle of ideas connected with theory of rough singular integrals. Partial differential equations form a basis for mathematicalmodeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations. Among the modern approaches to the study of these equations is the application of singular integrals and the associated powerful techniques of harmonic analysis. ***