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

Abstract

The asymptotics is examined for solutions to the spectral problem for the Laplace operator in a d d -dimensional thin, of diameter O ( h ) O(h) , spindle-shaped domain Ω h \Omega ^h with the Dirichlet condition on small, of size h ≪ 1 h\ll 1 , terminal zones Γ ± h \Gamma ^h_\pm and the Neumann condition on the remaining part of the boundary ∂ Ω h \partial \Omega ^h . In the limit as h → + 0 h\rightarrow +0 , an ordinary differential equation on the axis ( − 1 , 1 ) ∋ z (-1,1)\ni z of the spindle arises with a coefficient degenerating at the points z = ± 1 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 d=3 it is necessary to construct boundary layers near the sets Γ ± h \Gamma ^h_\pm and in the case of d = 2 d=2 it is necessary to deal with selfadjoint extensions of the differential operator. The extension parameters depend linearly on ln ⁡ h \ln h so that its eigenvalues are analytic functions in the variable 1 / | ln ⁡ h | 1/|\ln h| . As a result, in all dimensions the one-dimensional model gets the power-law accuracy O ( h δ d ) O(h^{\delta _d}) with an exponent δ d > 0 \delta _d>0 . First (the smallest) eigenvalues, positive in Ω h \Omega ^h and null in ( − 1 , 1 ) (-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.