Homogenization of elliptic equations in thin cylinders and related qualitative problems

Abstract: The thesis consists of four closely related papers (A, B, C and D). Paper A concerns the study of the behaviour at infinity of solutions to second order elliptic equation with first order terms stated in a half-cylinder. Neumann boundary condition is imposed on the lateral boundary of the cylinder, while on the base we assign the Dirichlet boundary condition. Under the assumption that the coefficients of the considered equation stabilize to a periodic regime exponentially, and the functions on the right-hand side decay sufficiently fast at infinity, we prove the existence of a bounded solution and its stabilization to a constant at the exponential rate at infinity. Also we provide a necessary and sufficient condition for the uniqueness of a bounded solution. Paper B is devoted to the homogenization of a stationary convection-diffusion equation in a thin cylinder being a union of two nonintersecting rods with a junction at the origin. It is assumed that each of these cylinders has a periodic microstructure,and that the microstructure period is of the same order as the cylinder diameter. Under natural assumptions on the data we construct and justify the asymptotic expansion of a solution which consists of the interior expansion and the boundary layer correctors arising both in the vicinity of the rod ends and the vicinity of the junction. Constructing boundary layers correctors relies essentially on the results of Paper A. Paper C deals with the asymptotic behaviour of the lower part of spectra of second-order self-adjoint elliptic operators with periodic rapidly oscillating coefficients under the assumption that the spectral density function changes sign. We study the Dirichlet problem in a regular bounded domain and show that the spectrum is discrete and consists of two sequences, tending to plus and minus infinity. It turns out that the asymptotic behaviour of eigenpairs depends essentially on whether the average of the spectral density function is equal to zero or not. We construct and justify the asymptotic expansion of eigenpairs in both cases. Paper D is aimed at homogenization of a spectral problem for a second-order self-adjoint elliptic operator stated in a thin cylinder with homogeneous Neumann boundary condition on the lateral boundary and Dirichlet boundary condition on the bases of the cylinder. It is assumed that the operator coefficients, as well as the spectral density function, are locally periodic in the axial direction of the cylinder, and that the spectral density function changes its sign. We show that the behaviour of the spectrum depends on whether the average of the density function is zero or not. In both cases we construct 1-dimensional effective spectral problem and prove the convergence of the spectra, for zero average densities the limit spectral problem being that for quadratic operator pencil. The results obtained in Paper A allow us to build the boundary layer correctors in the vicinity of the cylinder bases and, as a result, essentially improve the asymptotics.

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