## Three-Term Asymptotics of the Spectrum of Self-Similar Fractal Drums

J. Math. Sci. Univ. Tokyo
Vol. 6 (1999), No. 1, Page 101--126.

Gerling, JÃ¼rgen ; Schmidt, Heinz-JÃ¼rgen
Three-Term Asymptotics of the Spectrum of Self-Similar Fractal Drums
In the present paper we consider the number $\cN_\Om(\la)$ of eigenvalues not exceeding $\la$ of the negative Laplacian with homogeneous {\sc Dirichlet} boundary conditions in a domain $\Om\subset\RR^n$ with fractal boundary $\partial \Om$. It is known that for $\la\to\infty$, $\cN_\Om(\la)=\cC_n|\Om|_n\la^{n/2}+O(\la^{D/2})$, where $D$ is the {\sc Minkowski} dimension of $\partial\Om$. For a certain class of domains with self--similar boundary, so-called "fractal drums", we obtain a second term of the form $-\cF(\ln\la)\,\la^{D/2}$ with a bounded periodic function $\cF$ and a third term. We investigate the function $\cF$ which contains a generalized {\sc Weierstrass} function with a self--similar fractal graph. Exact estimates for the {\sc Minkowski} dimension for this graph will be presented.