Research output: Contribution to journal › Article
Asymptotics of the eigenvalues of boundary value problems for the laplace operator in a three-dimensional domain with a thin closed tube. / Nazarov, S.A.
In: Transactions of the Moscow Mathematical Society, 2015, p. 1-53.Research output: Contribution to journal › Article
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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 -
ID: 4011897