Abstract
Abstract: If one knows geometric properties of a domain, such as its area or perimeter, then what inequalities can one deduce on the eigenvalues of the Laplacian for that domain? For example, the fundamental tone (first eigenvalue) and spectral functionals such as the spectral zeta function and heat trace have long been known to be extremal when the domain is a ball, provided the volume of the domain is fixed. We prove complementary bounds in the opposite direction (again sharp for the ball) by introducing an additional geometric quantity that measures the ”deviation of the domain from roundness”. An intriguing role in the proof is played by volume-preserving diffeomorphisms that are not the identity map. [Joint work with B. Siudeja, U. of Oregon]
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