**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
- 28A80 Fractals, See also {58Fxx}

**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*

**Upload:** 1999-01-28

**Update:** 2000-01-20

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