The distribution of eigenfrequencies of anisotropic fractal drums

Preprint series: 98-21, Analysis

The paper is published: J. London Math. Soc., 60 (1), 224-236, 1999.

MSC:
46E35 Sobolev spaces and other spaces of smooth'' functions, embedding theorems, trace theorems
35P15 Estimation of eigenvalues, upper and lower bounds

Abstract: Let $\Gamma$ be an anisotropic fractal. The aim of the paper is to investigate the distribution of
the eigenvalues of the fractal differential operator $$(-\Delta )^{-1}\, \circ tr ^{\Gamma}$$ acting in
the classical Sobolev space $\stackrel{\circ}{W}\!\!{}_{2}^{1}(\Omega )$ where $\Omega$ is a bounded
$C^{\infty}$ domain in the plane ${\mathbb R} ^2$ with $\Gamma\subset\Omega$. Here $-\Delta$ is the
Dirichlet Laplacian in $\Omega$ and $tr ^{\Gamma}$ is closely related to the trace operator $tr _{\Gamma}$.

Keywords: spectral theory, fractals