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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
##### Keywords:
Neumann problem; thin set; eigenvalue; Laplace-Beltrami operator
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##### References:
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