Universal and Isoperimetric Inequalities for the Eigenvalues of the Fixed Membrane Problem with Lotfi Hermi, FIU Mathematics and Statistics
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Thursday, November 10, 2022 at 3:45pm to 4:45pm
DM - Deuxieme Maison, 409A
11200 SW 8th ST 33199, Deuxieme Maison, Miami, Florida 33199
Abstract The eigenvalues of the fixed membrane problem (corresponding to the eigenvalues of the Dirichlet problem on a domain in Euclidean space) satisfy tight universal and isoperimetric inequalities. In this colloquium, designed to be accessible to grad students, I will survey several problems of interest, and show how the tools of integral transforms can be used in combination with the tools of convex analysis, and the Rayleigh-Ritz principle, to show universal Weyl-sharp inequalities for ratios and averages of these eigenvalues. For wedge-like membranes, and domains in a convex cone, one can in fact improve on the classical Faber-Krahn inequality and prove weighted isoperimetric inequalities which improve on classical results by recourse to the Weinstein interpretation in fractional space, and by introducing weighted forms of rearrangement à la Talenti.
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Meeting ID: 986 4230 7589
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