TY - JOUR

T1 - Asymptotics of the eigenvalues of boundary value problems for the laplace operator in a three-dimensional domain with a thin closed tube

AU - Nazarov, S.A.

PY - 2015

Y1 - 2015

N2 - © 2015 S. A. Nazarov.We construct and justify asymptotic representations for the eigenvalues and eigenfunctions of boundary value problems for the Laplace operator in a three-dimensional domain (formula presented) with a thin singular set Γε lying in the cε-neighborhood of a simple smooth closed contour Γ. We consider the Dirichlet problem, a mixed boundary value problem with the Neumann conditions on ∂Γε, and also a spectral problem with lumped masses on Γε. The asymptotic representations are of diverse character: we find an asymptotic series in powers of the parameter |ln ε|−1 or ε. The most comprehensive and complicated analysis is presented for the lumped mass problem; namely, we sum the series in powers of |ln ε|−1 and obtain an asymptotic expansion with the leading term holomorphically depending on |ln ε|−1 and with the remainder O(εδ), δ ∈ (0, 1). The main role in asymptotic formulas is played by solutions of the Dirichlet problem in Ω\Γ with logarithmic singularities distributed along the contour Γ.

AB - © 2015 S. A. Nazarov.We construct and justify asymptotic representations for the eigenvalues and eigenfunctions of boundary value problems for the Laplace operator in a three-dimensional domain (formula presented) with a thin singular set Γε lying in the cε-neighborhood of a simple smooth closed contour Γ. We consider the Dirichlet problem, a mixed boundary value problem with the Neumann conditions on ∂Γε, and also a spectral problem with lumped masses on Γε. The asymptotic representations are of diverse character: we find an asymptotic series in powers of the parameter |ln ε|−1 or ε. The most comprehensive and complicated analysis is presented for the lumped mass problem; namely, we sum the series in powers of |ln ε|−1 and obtain an asymptotic expansion with the leading term holomorphically depending on |ln ε|−1 and with the remainder O(εδ), δ ∈ (0, 1). The main role in asymptotic formulas is played by solutions of the Dirichlet problem in Ω\Γ with logarithmic singularities distributed along the contour Γ.

U2 - 10.1090/mosc/243

DO - 10.1090/mosc/243

M3 - Article

SP - 1

EP - 53

JO - Transactions of the Moscow Mathematical Society

JF - Transactions of the Moscow Mathematical Society

SN - 0077-1554

ER -