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 Izv. RAN. Ser. Mat., 1993, Volume 57, Issue 1, Pages 202–239 (Mi izv897)

Asymptotic of the solution of a boundary value problem in a thin cylinder with nonsmooth lateral surface

S. A. Nazarov

Abstract: The mixed boundary value problem is considered for a selfadjoint elliptic second order equation in a three-dimensional cylinder $Q_\varepsilon$ of small height $\varepsilon$, with Dirichlet conditions on the lateral surface and Neumann conditions on the bases. The cross-section $\Omega$ of the cylinder has a corner point at 0. The full asymptotic expansion of the solution in a series of powers of the small parameter $\varepsilon$ is derived. In contrast to the iterative processes for a smooth boundary $\partial\Omega$, here there arises an additional (corner) boundary layer in the neighborhood of 0. This layer is described by means of the solutions of the boundary value problem in the domain $t=K\times(-\frac12, \frac12)$, where $K$ is a plane angle. The solvability of the problem is investigated in some Hilbert spaces of functions with weighted norms, and asymptotic representations of the solutions at infinity are established. The construction of the asymptotics of the solution with respect to $\varepsilon$ is based on the method of redistribution of residuals between the right-hand sides of the limiting problems.

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English version:
Russian Academy of Sciences. Izvestiya Mathematics, 1994, 42:1, 183–217

Bibliographic databases:

UDC: 517.946
MSC: 35J25, 35B25, 35C20

Citation: S. A. Nazarov, “Asymptotic of the solution of a boundary value problem in a thin cylinder with nonsmooth lateral surface”, Izv. RAN. Ser. Mat., 57:1 (1993), 202–239; Russian Acad. Sci. Izv. Math., 42:1 (1994), 183–217

Citation in format AMSBIB
\Bibitem{Naz93} \by S.~A.~Nazarov \paper Asymptotic of the solution of a boundary value problem in a thin cylinder with nonsmooth lateral surface \jour Izv. RAN. Ser. Mat. \yr 1993 \vol 57 \issue 1 \pages 202--239 \mathnet{http://mi.mathnet.ru/izv897} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1220589} \zmath{https://zbmath.org/?q=an:0807.35031} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?1994IzMat..42..183N} \transl \jour Russian Acad. Sci. Izv. Math. \yr 1994 \vol 42 \issue 1 \pages 183--217 \crossref{https://doi.org/10.1070/IM1994v042n01ABEH001531} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1994NH32100011} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. D. B. Rokhlin, “Impact on a planar body floating on the surface of a thin layer of an inviscid incompressible fluid”, Comput. Math. Math. Phys., 38:8 (1998), 1312–1322
2. S. A. Nazarov, M. Specovius-Neugebauer, “Artificial boundary conditions providing superpolynomial error estimates for the Neumann problem in a layered domain”, Comput. Math. Math. Phys., 43:10 (2003), 1418–1429
3. Sergueı̈ A Nazarov, Gudrun Thäter, “Asymptotics at infinity of solutions to the Neumann problem in a sieve-type layer”, Comptes Rendus Mécanique, 331:1 (2003), 85
4. S. A. Nazarov, “Concentration of trapped modes in problems of the linearized theory of water waves”, Sb. Math., 199:12 (2008), 1783–1807
5. Bunoiu R. Cardone G. Nazarov S.A., “Scalar Boundary Value Problems on Junctions of Thin Rods and Plates”, ESAIM-Math. Model. Numer. Anal.-Model. Math. Anal. Numer., 48:5 (2014), 1495–1528
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