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Asymptotic expansion of eigenelements of the Laplace operator in a domain with a large number of ‘light’ concentrated masses sparsely situated on the boundary. Two-dimensional case. (English. Russian original) Zbl 1202.35143

Trans. Mosc. Math. Soc. 2009, 71-134 (2009); translation from Tr. Mosk. Mat. O.-va 70, 102-182 (2009).
Summary: This paper looks at eigenoscillations of a membrane containing a large number of concentrated masses on the boundary. The asymptotic behaviour of the frequencies of eigenoscillations is studied when a small parameter characterizing the diameter and density of the concentrated masses tends to zero. Asymptotic expansions of eigenelements of the corresponding problems are constructed and the expansions are accurately substantiated. The case where the diameter of the masses is much smaller than the distance between them is investigated under the assumption that the limit boundary condition is still a Dirichlet condition.

MSC:

35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs
35J25 Boundary value problems for second-order elliptic equations
35B25 Singular perturbations in context of PDEs
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35B40 Asymptotic behavior of solutions to PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
74K15 Membranes
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