Steklov spectral problems in a set with a thin toroidal hole
Valeria Chiadò Piat, С. А. Назаров · Partial Differential Equations in Applied Mathematics · 2020
The paper concerns the Steklov spectral problem for the Laplace operator, and some variants in a 3-dimensional bounded domain, with a cavity Γε having 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.