On the eigenfunctions and the first eigenvalue of fractal drums

Preprint series: 99-14, Analysis

MSC:
46E35 Sobolev spaces and other spaces of smooth'' functions, embedding theorems, trace theorems
35P20 Asymptotic distribution of eigenvalues and eigenfunctions for PDO
Abstract: The paper deals with the smoothness of eigenfunctions and the simplicity of the first eigenvalue of $B = ( - \Delta )^{- 1} \circ \mu$, where $-\Delta$ is the Dirichlet Laplacian in a bounded $C^\infty$ domain in ${\Bbb{R}}^n$ and $\mu$ is a fractal measure.