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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.

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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