Steklov spectral problems in a set with a thin toroidal hole

• Main Authors: V. Chiadò Piat, S.A. Nazarov
• Format: Article
• Language: English
• Published: Elsevier 2020-09-01
• Series: Partial Differential Equations in Applied Mathematics
• Subjects: Steklov spectral problems; Singularly perturbed domains; Thin torus excluded; Asymptotics of eigenfunctions and eigenvalues; Multiplicity; Series in powers of logarithms
• Online Access: http://www.sciencedirect.com/science/article/pii/S2666818120300073
• ISSN: 2666-8181

The paper concerns the Steklov spectral problem for the Laplace operator, and some variants in a 3-dimensional bounded domain, with a cavity Γεhaving the shape of a thin toroidal set, with a constant cross-section of diameter ε≪1. We construct the main terms of the asymptotic expansion of the eigenvalues in terms of real-analytic functions of the variable |lnε|−1, and we prove that the relative asymptotic error is of much smaller order O(ε|lnε|)as ε→0+. The asymptotic analysis involves eigenvalues and eigenfunctions of a certain integral operator on the smooth curve Γ, the axis of the cavity Γε.