Mathematical Sciences: Maximum Principles and Dilation Invariant Estimates for Sobolev and Dirichlet Problems

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9401354 Verchota This project in mathematical analysis focuses on the application of methods of harmonic analysis to the study of partial differential equations. Work will be done on the formation of estimates of solutions to boundary value problems of elliptic operators with constant coefficients. The tools employed include singular integrals and Sobolev and Hardy spaces on lower dimensional Lipschitz surfaces. The lower dimensional theory, which is just developing, seeks to yield new maximum principles for higher order equations and systems. One of the underlying goals of the research is that of understanding which properties of dilation invariant solutions might be considered operator independent. Classical theory fails to distinguish those properties of solutions that are operator dependent from those that are domain dependent. Boundary value problems such as the Dirichlet problem are to be generalized in an algebraic way to higher order operators and systems that may not conform to intuition on how values at the boundary should control values inside a domain. Partial differential equations form a basis for mathematical modeling 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. ***