Singularly Perturbed Spectral Problems in a Thin Cylinder with Fourier Conditions on its Bases

Andrey Piatnitski

The Arctic University of Norway, Campus in Narvik, P.O. Box 385, N-8505 Narvik,
Norway Institute for Information Transmission Problems RAS, Bolshoi Karetnyi, 19, Moscow, 127051, Russia
E-mail: apiatnitski@gmail.com

Volodymyr Rybalko

B. Verkin Institute for Low Temperature Physics and Engineering of the National
Academy of Sciences of Ukraine, 47 Nauky Ave., Kharkiv, 61103, Ukraine
E-mail: vrybalko@ilt.kharkov.ua

Received April 1, 2019

Abstract

The paper deals with the bottom of the spectrum of a singularly perturbed
second order elliptic operator defined in a thin cylinder and having
locally periodic coefficients in the longitudinal direction. We impose a homogeneous Neumann boundary condition on the lateral surface of the cylinder
and a generic homogeneous Fourier condition at its bases. We then show
that the asymptotic behavior of the principal eigenpair can be characterized
in terms of the limit one-dimensional problem for the effective Hamilton-Jacobi equation with the effective boundary conditions. In order to construct boundary layer correctors we study a Steklov type spectral problem
in a semi-infinite cylinder (these results are of independent interest). Under a structure assumption on the effective problem leading to localization (in certain sense) of eigenfunctions inside the cylinder we prove a two-term asymptotic formula for the first and higher order eigenvalues.