Asymptotics of eigenvalues of the Laplacian with small spherical Robin boundary
Let \(\Omega\) be a bounded domain in \(\mathbb{R}^ 3\) with \(C^ \infty\) boundary \(\partial \Omega\). Let \(\omega \in \Omega\) be a fixed point and \(B (\varepsilon, \omega) = \{x \in \Omega : | x - \omega | < \varepsilon\}\); let \(\Omega_ \varepsilon = \Omega \backslash B (\varepsilon, \omega)\). The author studies the eigenvalue problem \[ - \Delta u(x) = \lambda u(x)\;(x \in \Omega_ \varepsilon), \quad u(x) |_{\partial \Omega} = 0, \quad u(x) + K \varepsilon^ \sigma D_ \nu u(x) = 0 \quad (x \in \partial B (\varepsilon, \omega)). \tag{1} \] Here \(K\) denotes a positive constant, \(\sigma\) is a real number, \(D_ \nu\) denotes the derivative along the exterior normal direction with respect to \(\Omega_ \varepsilon\). Let \(\mu_ j (\varepsilon) > 0\) be the \(j\)-th eigenvalue of (1) and let \(\mu_ j\) be the \(j\)-th eigenvalue of the problem \[ -\Delta u(x) = \lambda u(x)\;(x \in \Omega), \quad u(x) = 0\;(x \in \partial \Omega). \tag{2} \] The author proves that if \(\mu_ j\) is simple then for any fixed \(s \in (0,1)\) \[ \begin{aligned} \mu_ j (\varepsilon) & = \mu_ j + P_ j \varepsilon + O (\varepsilon^{2- s})\;(\sigma \geq 2); \mid \mu_ j (\varepsilon) = \mu_ j + P_ j \varepsilon + O (\varepsilon^ \sigma)\;(1 < \sigma < 2), \\ \mu_ j (\varepsilon) & = \mu_ j + (1 + K)^{-1} P_ j \varepsilon + O (\varepsilon^{2-s})\;(\sigma = 1) \end{aligned} \] where \(P_ j = 4 \pi \varphi_ j (\omega)^ 2\) and \(\varphi_ j\) is the \(L^ 2\)-normalized eigenfunction associated with \(\mu_ j\). The asymptotics for \(\sigma < 1\) is also obtained. This asymptotic is different on the intervals \([0,1)\), \((- 1/2,0)\), \((- 2,- 1/2]\), \((- \infty, - 2]\).
scientific article; zbMATH DE number 591440