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 タイトル: Three-Term Asymptotics of the Spectrum of Self-Similar Fractal Drums 著者: Gerling, Jurgen 発行日: 1999年 出版者: Graduate School of Mathematical Sciences, The University of Tokyo 掲載誌情報: Journal of Mathematical Sciences, The University of Tokyo. Vol. 6 (1999), No. 1, Page 101-126 抄録: 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. URI: http://hdl.handle.net/2261/1247 ISSN: 13405705 出現カテゴリ: Journal of Mathematical Sciences, the University of TokyoJournal of Mathematical Sciences, the University of Tokyo

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