Can one see the fundamental frequency of a drum?

Details Can one see the fundamental frequency of a drum? Can one see the fundamental frequency of a drum?

2005-06-09

arxiv.org/abs/math/0506181v3

Mathematics

Spectral Theory

18 pages, some misprints corrected

Scientific paper

10.1007/s11005-005-0010-1

We establish two-sided estimates for the fundamental frequency (the lowest eigenvalue) of the Laplacian in an open subset G of R^n with the Dirichlet boundary condition. This is done in terms of the interior capacitary radius of G which is defined as the maximal possible radius of a ball B which has a negligible intersection with the complement of G. Here negligibility of a subset F in B means that the Wiener capacity of F does not exceed gamma times the capacity of B, where gamma is an arbitrarily fixed constant between 0 and 1. We provide explicit values of constants in the two-sided estimates.

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