Abstract:

The fundamental frequency of a drum is given by the first eigenvalue
of the Laplacian with Dirichlet boundary data. Rayleigh conjectured
in 1877 and Faber and Krahn showed in 1923 that among all domains of
given volume, the round ball has the lowest first eigenvalue. Their
method uses the isoperimetric inequality, which says that among
domains with given volume, the ball has least surface area. A similar
eigenvalue bound for surfaces depending on an isoperimetric constant
was given by Cheeger in 1968. This gives a sharp, geometric constant
for the Poincare inequality.

For non-round domains, a better inequality can be given in terms of
the radius of the largest ball contained in the domain. We present a
simple maximum principle argument of Li and Yau that generalizes to
manifolds.