The paper concerns the Steklov spectral problem for the Laplace operator, and some variants in a 3-dimensional bounded domain, with a cavity $Gamma_e$ having the shape of a thin toroidal set, with a constant cross-section of diameter $ell 1$. We construct the main terms of the asymptotic expansion of the eigenvalues in terms of real-analytic functions of the variable $|lne|^{-1}$, and we prove that the relative asymptotic error is of much smaller order $O(e|ln e|)$ as $e o 0^+$. The asymptotic analysis involves eigenvalues and eigenfunctions of a certain integral operator on the smooth curve $Gamma$, the axis of the cavity $Gamma_e$.

Steklov spectral problems in a set with a thin toroidal hole / Chiado' Piat, V.; A: Nazarov, S.. - In: PARTIAL DIFFERENTIAL EQUATIONS IN APPLIED MATHEMATICS. - ISSN 2666-8181. - ELETTRONICO. - 1:(2020), p. 100007. [10.1016/j.padiff.2020.100007]

### Steklov spectral problems in a set with a thin toroidal hole

#### Abstract

The paper concerns the Steklov spectral problem for the Laplace operator, and some variants in a 3-dimensional bounded domain, with a cavity $Gamma_e$ having the shape of a thin toroidal set, with a constant cross-section of diameter $ell 1$. We construct the main terms of the asymptotic expansion of the eigenvalues in terms of real-analytic functions of the variable $|lne|^{-1}$, and we prove that the relative asymptotic error is of much smaller order $O(e|ln e|)$ as $e o 0^+$. The asymptotic analysis involves eigenvalues and eigenfunctions of a certain integral operator on the smooth curve $Gamma$, the axis of the cavity $Gamma_e$.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11583/2854917