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Spectra of domains with small spherical Neumann boundary. (English) Zbl 0541.35061

Let \(\Omega\) be a bounded domain in \(R^ 2\) with \(C^{\infty}\) boundary \(\gamma\) and let \(B_{\epsilon}\) be the \(\epsilon\)-ball whose center is \(\tilde w\in \Omega\). We put \(\Omega_{\epsilon}=\Omega \backslash \bar B_{\epsilon}\). We consider the following eigenvalue problem: \[ -\Delta u(x)=\lambda(\epsilon)\quad u(x),\quad x\in \Omega_{\epsilon},\quad u(x)=0,\quad x\in \gamma,\quad \partial u(x)/\partial \nu =0,\quad x\in \partial B_{\epsilon}, \] where \(\partial /\partial \nu\) denotes the derivative along the inner normal vector at x with respect to the domain \(\Omega_{\epsilon}\). Let \(0<\mu_ 1(\epsilon)\leq \mu_ 2(\epsilon)\leq..\). be the eigenvalues of the above problem. And let \(0<\mu_ 1\leq \mu_ 2\leq..\). be the eigenvalue of -\(\Delta\) in \(\Omega\) under the Dirichlet condition on \(\gamma\). We arrange them repeatedly according to their multiplicities. We prove the following Theorem: for fixed j assume that \(\mu_ j\) is a simple eigenvalue. Then \[ \mu_ j(\epsilon)=\mu_ j-(2\pi | \text{grad} \phi_ j(\tilde w)|^ 2-\pi \mu_ j\phi_ j(\tilde w)^ 2)\epsilon^ 2+O(\epsilon^ 3| \log \epsilon |^ 2) \] holds as \(\epsilon\) tends to zero, where \(\phi_ j(x)\) denotes the eigenfunction associated with \(\mu_ j\) satisying \(\int_{\Omega}\phi_ j(x)^ 2\quad dx=1.\)

MSC:

35P20 Asymptotic distributions of eigenvalues in context of PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B20 Perturbations in context of PDEs