Asymptotics of the spectrum of the mixed boundary value problem for the Laplace operator in a thin spindle-shaped domain

Nazarov, S.; Taskinen, J.

doi:10.1090/spmj/1701

Asymptotics of the spectrum of the mixed boundary value problem for the Laplace operator in a thin spindle-shaped domain
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by S. A. Nazarov and J. Taskinen
Translated by: S. A. Nazarov

St. Petersburg Math. J. 33 (2022), 283-325

DOI: https://doi.org/10.1090/spmj/1701

Published electronically: March 4, 2022

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Abstract:

The asymptotics is examined for solutions to the spectral problem for the Laplace operator in a $d$-dimensional thin, of diameter $O(h)$, spindle-shaped domain $\Omega ^h$ with the Dirichlet condition on small, of size $h\ll 1$, terminal zones $\Gamma ^h_\pm$ and the Neumann condition on the remaining part of the boundary $\partial \Omega ^h$. In the limit as $h\rightarrow +0$, an ordinary differential equation on the axis $(-1,1)\ni z$ of the spindle arises with a coefficient degenerating at the points $z=\pm 1$ and moreover, without any boundary condition because the requirement on the boundedness of eigenfunctions makes the limit spectral problem well-posed. Error estimates are derived for the one-dimensional model but in the case of $d=3$ it is necessary to construct boundary layers near the sets $\Gamma ^h_\pm$ and in the case of $d=2$ it is necessary to deal with selfadjoint extensions of the differential operator. The extension parameters depend linearly on $\ln h$ so that its eigenvalues are analytic functions in the variable $1/|\ln h|$. As a result, in all dimensions the one-dimensional model gets the power-law accuracy $O(h^{\delta _d})$ with an exponent $\delta _d>0$. First (the smallest) eigenvalues, positive in $\Omega ^h$ and null in $(-1,1)$, require individual treatment. Also, infinite asymptotic series are discussed, as well as the static problem (without the spectral parameter) and related shapes of thin domains.

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