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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].

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