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Asymptotic property of an eigenfunction of the Laplacian under singular variation of domains - the Neumann condition -. (English) Zbl 0579.35065

Let \(\Omega\) be a bounded domain in \({\mathbb{R}}_ 2\) with smooth boundary \(\gamma\). Let \(B_{\epsilon}\) be the \(\epsilon\) disk whose center is \(w\in \Omega\). The eigenvalue problem \[ -\Delta_ xu(x)=\lambda (\epsilon)u(x)\quad x\in \Omega \setminus B_{\epsilon};\quad u(x)=0,\quad x\in \gamma;\quad \frac{\partial u}{\partial \nu}(x)=0\quad x\in B_{\epsilon} \] is studied. Let \(0<\mu_ 1(\epsilon)\leq \mu_ 2(\epsilon)\leq...\), be the eigenvalues of the above mentioned problem. Denote by \(\phi_ j(\epsilon)\) the eigenfunction associated with \(\mu_ j(\epsilon)\), \(j=1,2,... \). The following problem is investigated: What can one say about asymptotic behaviour of \(\phi_ j(\epsilon)\) as \(\epsilon\) tends to zero? Two theorems are proved on the asymptotic behaviour of the Laplacian under singular variation of domains.
Reviewer: I.Ecsedi

MSC:

35P15 Estimates of eigenvalues in context of PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35B20 Perturbations in context of PDEs
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