Mathematical Sciences: Isoperimetric Inequalities for Eigenvalues in Mathematical Physics and Geometry

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Ashbaugh DMS-9500968 The research that Ashbaugh will conduct concerns finding optimal bounds for the eigenvalues of the classical partial differential operators arising in mathematical physics and differential geometry. Specific examples include Rayleigh's conjecture for the lowest eigenvalue of the clamped plate for all dimensions, and the Payne-Polya-Weinberger conjecture on the ratio of eigenvalues of the Dirichlet Laplacian on an arbitrary bounded domain in Euclidean n-space. 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.