English

Can one see the fundamental frequency of a drum?

Spectral Theory 2009-11-11 v3 Mathematical Physics math.MP

Abstract

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.

Keywords

Cite

@article{arxiv.math/0506181,
  title  = {Can one see the fundamental frequency of a drum?},
  author = {Vladimir Maz'ya and Mikhail Shubin},
  journal= {arXiv preprint arXiv:math/0506181},
  year   = {2009}
}

Comments

18 pages, some misprints corrected