Asymptotics of the eigenvalues of boundary value problems for the Laplace operator in a three-dimensional domain with a thin closed tube

Abstract:

We construct and justify asymptotic representations for the eigenvalues and eigenfunctions of boundary value problems for the Laplace operator in a three-dimensional domain $\Omega (\varepsilon )=\Omega \setminus \bar {\Gamma }_\varepsilon$ with a thin singular set $\Gamma _\varepsilon$ lying in the $c\varepsilon$-neighborhood of a simple smooth closed contour $\Gamma$. We consider the Dirichlet problem, a mixed boundary value problem with the Neumann conditions on $\partial \Gamma _\varepsilon$, and also a spectral problem with lumped masses on $\Gamma _\varepsilon$. The asymptotic representations are of diverse character: we find an asymptotic series in powers of the parameter $|{\ln \varepsilon }|^{-1}$ or $\varepsilon$. The most comprehensive and complicated analysis is presented for the lumped mass problem; namely, we sum the series in powers of $|{\ln \varepsilon }|^{-1}$ and obtain an asymptotic expansion with the leading term holomorphically depending on $|{\ln \varepsilon }|^{-1}$ and with the remainder $O(\varepsilon ^\delta )$, $\delta \in (0,1)$. The main role in asymptotic formulas is played by solutions of the Dirichlet problem in $\Omega \setminus \Gamma$ with logarithmic singularities distributed along the contour $\Gamma$.