Spectra of the Laplacian with small Robin conditional boundary (Q1924881)

Let \(M\) be a bounded domain in \(\mathbb{R}^4\) with smooth boundary \(\partial M\). Let \(B_\varepsilon\) be the ball of radius \(\varepsilon\) with the center \(\widetilde{w}\), \(M_\varepsilon=M\setminus \overline{B}_\varepsilon\). Consider the following eigenvalue problem: \[ -\Delta u(x)=\lambda u(x), \quad x\in M_\varepsilon, \qquad u(x)=0, \quad x\in\partial M \] \[ u(x)+k\varepsilon^\sigma {\partial\over{\partial\nu_x}} u(x)=0, \qquad x\in\partial B_\varepsilon. \] Let \(\mu_j(\varepsilon)\) be the \(j\)th eigenvalue of the above problem, and \(\mu_j\) be the \(j\)th eigenvalue of the problem: \[ -\Delta u(x)=\lambda u(x), \quad x\in M, \qquad u(x)=0, \quad x\in\partial M. \] Let \(\varphi_j(x)\) be the normalized eigenfunction associated with \(\mu_j\). We prove the following: Assume that \(\mu_j\) is a simple eigenvalue. Then, \[ \mu_j(\varepsilon)- \mu_j=2\pi^2k^{-1} \varepsilon^{3-\sigma} \varphi_j(\widetilde{w})^2+ O(\varepsilon^{4-2\sigma}+ \varepsilon^{3-\sigma} (\varepsilon^{(1/2)+\theta}+ \varepsilon^{\sigma+\theta})) \] for some \(\theta>0\) as \(\varepsilon\) tends to zero.