Splitting a multiple eigenvalue in the boundary value problem for a membrane clamped on a small part of the boundary
Let \(\lambda_0\) be an \(m\)-multiple eigenvalue of the Neumann problem for the Laplace operator in a bounded connected domain \(\Omega\) of \(\mathbb{R}^2\) with boundary of class \(C^\infty\) and \(\varphi_i\) \((i = 1, \ldots, m)\) the corresponding eigenfunctions; then there exists \(\varepsilon_m > 0\) such that for \(0 < \varepsilon < \varepsilon_m\) there exist \(m\) simple eigenvalues \(\lambda_{\varepsilon,i}\) of the problem \((\Delta + \lambda_\varepsilon) \varphi = 0\) in \(\Omega\), \(\partial_n \varphi = 0\) on \(\partial \Omega \backslash \overline {\omega_\varepsilon}\), \(\varphi = 0\) on \(\omega_\varepsilon\), where \(\omega_\varepsilon\) is an open connected subset of \(\partial \Omega\) with length \(\varepsilon\), and \(\lambda_{\varepsilon,i} \to \lambda_0\) as \(\varepsilon \to 0\). Moreover the author shows the form of asymptotic expansions of the eigenvalues \(\lambda_{\varepsilon,i}\) and of the corresponding eigenfunctions if \(\sum^m_{i = 1} |\varphi_i (0) |\neq 0\) and if \(\sum^m_{i = 1} |\varphi_i (0) |= 0\).
Neumann problem for the Laplace operator
scientific article; zbMATH DE number 641339