We consider a two-dimensional spectral problem of Steklov type for the Laplace operator in a domain divided into two parts by a perforated partition with a periodic microstructure. The Steklov boundary condition is imposed on the lateral sides of the perforation, the Neumann condition on the remaining part of the boundary, and the Dirichlet and Neumann conditions on the outer boundary of the domain. We construct and justify two-term asymptotic expressions for the eigenvalues of this problem. We also construct a two-term asymptotic formula for the corresponding eigenfunctions.
We consider the degenerate problem on low-frequency oscillations of a heavy viscous incompressible fluid in a vessel with free surface of inhomogeneous nonperiodic microstructure. For the obtained quadratic operator pencil we construct the limit (homogenized) pencil which turns out to be degenerate. We prove a homogenization theorem. Bibliography: 40 titles. Illustration: 2 figures.
Dedicated to Nina Nikolaevna UraltsevaA scalar analog of linear hydrodynamics was first described in [1] in detail. Such problems with surfaces of microinhomogeneous periodic structure were considered in [2]-[4], where homogenization of an operator pencil was studied. This study was based on homogenization methods (cf., for example, [5]-[13] and the references therein), asymptotic analysis, and matching asymptotic expansions (cf., for example, [14]). were used. Owing to this approach, it becomes possible to construct effective models of inhomogeneous media or study singular perturbations of problems without microinhomogeneities; moreover, these methods provide rigorous justifications of constructions and accurate proofs. In this direction, various problems with singular perturbation were considered, in particular, perturbation of geometry (cf., for example, [7,12], [15]-[17]), coefficients (cf. [18, 19]), and the type of boundary condition (cf., for example, [20]-[27] and [13]). Similar methods were used in hydrodynamics for studying strongly inhomogeneous fluids with different rheology (cf., for example, [28]-[39]).In this paper, we study a plane analog of the problem on low-frequency oscillations of a heavy viscous incompressible fluid in a vessel with free surface covered by a perforated cap with nonperiodic structure (cf. Figure 1).
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