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On the spectrum of the Steklov problem in peak-shaped domains. (English) Zbl 1183.35212
Uraltseva, N.N.(ed.), Proceedings of the St. Petersburg Mathematical Society. Vol. XIV. Transl. from the Russian by Tamara Rozhkovskaya. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4802-9/hbk). Translations. Series 2. American Mathematical Society 228, 79-131 (2009).
The author gives a detailed study of the spectrum of the Steklov spectral problems $\begin{cases}-\Delta_xu^0(x)=0,\,\,\,x\in\Omega\\ \partial_Nu^0(x)=\lambda^0u^0(x),\,\,\,x\in\partial\Omega\setminus{\mathcal O}\end{cases}$ and $\begin{cases}-\Delta_xu^{\varepsilon}(x)=0,\,\,\,x\in \Omega(\varepsilon)\\ \partial_Nu^{\varepsilon}(x)=\lambda^{\varepsilon}u^{\varepsilon}(x),\,\,\,x\in\Sigma(\varepsilon),\end{cases}$ where $$\Omega =\{x\in {\mathbb{R}}^n,\,\,x=(x',x_n),\,\,x'=(x_1,\ldots,x_{n-1}),\,\,x_n>0,\,\,x_n^{-2m}x'\in\omega\}$$ is bounded by an $$(n-1)$$-dimensional surface $$\partial \Omega$$ which is Lipschitz everywhere except for the origin $${\mathcal O}$$, $$m>1/2$$ is the cusp exponent and $$\omega$$ is a domain in the space $${\mathbb{R}}^{n-1}$$ with $$(n-2)$$-dimensional Lipschitz boundary $$\partial \omega$$ and compact closure $$\bar \omega=\omega\cup\partial \omega$$, $$\partial_N$$ is the derivative along the outward normal and $$\lambda^0,\,\,\lambda^{\varepsilon}$$ are spectral parameters, $$\Omega(\varepsilon)$$ is the peak-shaped domain $$\Omega$$ with “snapped-off” tip $$\Pi(\varepsilon)=\{x\in\Omega\cap {\mathcal U},\,\,x_n<\varepsilon\}$$ given by $$\Omega(\varepsilon)=(\Omega\setminus{\mathcal U})\cup\{x\in\Omega\cap{\mathcal U}:\,\,x_n>\varepsilon\}$$, with $${\mathcal U}=B_R^{n-1}\times (-d,d)$$, $$d>0$$ and $$B_R^{n-1}=\{x'\in {\mathbb{R}}^{n-1},\,\,\,|x'|<R\}$$, $$\Sigma(\varepsilon)$$ is $$\partial\Omega(\varepsilon)$$ or $$\partial\Omega(\varepsilon)\setminus\overline{\omega(\varepsilon)}$$. In particular the author describes the continuous spectrum for $$m\geq 1$$ and study the asymptotic behavior of eigenvalues in the domain $$\Omega(\varepsilon)$$ as $$\varepsilon\to +0$$.
For the entire collection see [Zbl 1179.00025].

##### MSC:
 35P20 Asymptotic distributions of eigenvalues in context of PDEs 35P05 General topics in linear spectral theory for PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation