Journal of Mathematical Physics, Analysis, Geometry
2019, vol. 15, No 2, pp. 256-277     ( to contents , go back )

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

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

Received April 1, 2019


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.

Mathematics Subject Classification 2000: 35B27,35P15, 35J25.
Key words: singularly perturbed operators, homogenization problems, eigenvalues, eigenfunctions, Fourier boundary conditions.

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