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 Γ ε 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 | ln ε | − 1 , and we prove that the relative asymptotic error is of much smaller order O ( ε | ln ε | ) 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 Γ ε .