Geodesic
Asymptotic analysis of boundary value and spectral problems in thin perforated regions with rapidly changing thickness and different limiting dimensions
Sbornik. Mathematics, Tome 203 (2012) no. 8, pp. 1169-1195

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Boundary value and spectral problems for an elliptic differential equation with rapidly oscillating coefficients in a thin perforated region with rapidly changing thickness are investigated. Descriptions of asymptotic algorithms for solutions of such problems in thin regions with different limiting dimensions are combined. For a mixed inhomogeneous boundary value problem a corrector is constructed and an asymptotic estimate in the corresponding Sobolev space is established. Asymptotic bounds for eigenvalues and eigenfunctions of the Neumann spectral problems are also found. Full asymptotic expansions for the eigenvalues and eigenfunctions are constructed under certain symmetry assumptions about the structure of the thin perforated region and the coefficients of the equations. Bibliography: 21 titles.
Keywords: asymptotic approximations and expansions, thin perforated regions, elliptic boundary value and spectral problems. 
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     author = {T. A. Mel'nik and A. V. Popov},
     title = {Asymptotic analysis of boundary value and spectral problems in thin perforated regions with rapidly changing thickness and different limiting dimensions},
     journal = {Sbornik. Mathematics},
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T. A. Mel'nik; A. V. Popov. Asymptotic analysis of boundary value and spectral problems in thin perforated regions with rapidly changing thickness and different limiting dimensions. Sbornik. Mathematics, Tome 203 (2012) no. 8, pp. 1169-1195. http://geodesic.mathdoc.fr/item/SM_2012_203_8_a4/