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On the eigenvalues of the Laplace operator on a thin set with Neumann boundary conditions. (English) Zbl 0865.35098
Let \(\Omega\) be an open bounded set of \(\mathbb{R}^n\), whose boundary is of class \(C^2\), and let \(\Omega (\varepsilon)\) be the set of \(x\in \mathbb{R}^n \setminus \Omega^c\) which lie at a distance less than \(\varepsilon\) of \(\Omega\). We show that the \(p\)-th eigenvalue \(\mu_p (\varepsilon)\) of \(-\Delta\) on \(\Omega (\varepsilon)\) converges to the \(p\)-th eigenvalue \(\lambda_p\) of the Laplace-Beltrami operator on \(\partial \Omega\). If \(\lambda_p\) is simple, we give the limit of \((\mu_p (\varepsilon)- \lambda_p)/ \varepsilon\) as \(\varepsilon\) tends to 0. These results are written in the language of analysts, and no knowledge of differential geometry is assumed.

MSC:
35P25 Scattering theory for PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35P20 Asymptotic distributions of eigenvalues in context of PDEs
Full Text: DOI
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