Eigenvalues of the Laplacian on domains with fractal boundary

Abstract

Consider the Laplacian operator on a bounded open domain in Euclidean space with Dirichlet boundary conditions. We show that for each number D with 1 < D < 2, there are two bounded open domains in R of the same area, with their boundaries having Minkowski dimension D, and having the same content, yet the secondary terms for the eigenvalue counts are not the same. This was shown earlier by Lapidus and the second author, but a possible countable set of exceptional dimensions D were excluded. Here we show that the earlier construction has no exceptions.