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Sibirskii Matematicheskii Zhurnal, 1993, Volume 34, Number 3, Pages 43–61
(Mi smj1651)
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This article is cited in 22 scientific papers (total in 22 papers)
Splitting a multiple eigenvalue in the boundary value problem for a membrane clamped on a small part of the boundary
R. R. Gadyl'shin
Abstract:
We prove that, under a singular perturbation of boundary conditions, a multiple eigenvalue $\lambda_0$ in the Neumann problem in a bounded connected domain $\Omega\subset\mathbb{R}^2$ with boundary $\Gamma_0\in C^\infty$ splits into the simple eigenvalues $\lambda_\varepsilon^{(i)}$ of the boundary value problem
\begin{gather*}
(\Delta+\lambda_{\varepsilon})\varphi_{\varepsilon}=0 \quad \text{for } x\in\Omega,
\\
\frac{\partial\varphi_{\varepsilon}}{\partial n}=0 \quad \text{on } \Gamma_0\setminus\overline{\omega}_{\varepsilon}, \quad \varphi_{\varepsilon}=0 \quad \text{on } \omega_{\varepsilon},
\end{gather*}
which possess distinct rates of convergence to $\lambda_0$. Here $\omega_{\varepsilon}$, is an open connected part of $\Gamma_0$ with length of order $\varepsilon$, $0<\varepsilon\ll 1$, and $n$ is the outward normal to $\Omega$.
Received: 24.09.1991
Citation:
R. R. Gadyl'shin, “Splitting a multiple eigenvalue in the boundary value problem for a membrane clamped on a small part of the boundary”, Sibirsk. Mat. Zh., 34:3 (1993), 43–61; Siberian Math. J., 34:3 (1993), 433–450
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https://www.mathnet.ru/eng/smj1651 https://www.mathnet.ru/eng/smj/v34/i3/p43
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