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 Algebra i Analiz: Year: Volume: Issue: Page: Find

 Algebra i Analiz, 2006, Volume 18, Issue 3, Pages 158–233 (Mi aa75)

Expository Surveys

Elastodynamics in domains with edges

S. I. Matyukevich, B. A. Plamenevskii

Saint-Petersburg State University

Abstract: Time-dependent boundary value problems with given displacements or stresses on the boundary of a domain are considered. The purpose is to describe the asymptotics of solutions near the edges of the boundary (including formulas for the “stress intensity factors”). The approach is based on various (energy and weighted) estimates of solutions. The weighted estimates in question are mixed in the sense that, in distinct zones, they involve derivatives of different orders. The method is implemented for problems in the cylinder $\mathbb D\times\mathbb R$, where $\mathbb D$ is an $m$-dimensional wedge, $m\ge 2$, and $\mathbb R$ is the time axis. For the cylinder $G\times\mathbb R$, where $G$ is a bounded domain with edges on the boundary, all the steps of the method are described except for the final one, which is related to the asymptotics itself. This step consists in compiling some known results of the theory of elliptic boundary value problems.

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English version:
St. Petersburg Mathematical Journal, 2007, 18:3, 459–510

Bibliographic databases:

MSC: 35L30, 35L35

Citation: S. I. Matyukevich, B. A. Plamenevskii, “Elastodynamics in domains with edges”, Algebra i Analiz, 18:3 (2006), 158–233; St. Petersburg Math. J., 18:3 (2007), 459–510

Citation in format AMSBIB
\Bibitem{MatPla06} \by S.~I.~Matyukevich, B.~A.~Plamenevskii \paper Elastodynamics in domains with edges \jour Algebra i Analiz \yr 2006 \vol 18 \issue 3 \pages 158--233 \mathnet{http://mi.mathnet.ru/aa75} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2255852} \zmath{https://zbmath.org/?q=an:05236247} \elib{http://elibrary.ru/item.asp?id=12894793} \transl \jour St. Petersburg Math. J. \yr 2007 \vol 18 \issue 3 \pages 459--510 \crossref{https://doi.org/10.1090/S1061-0022-07-00957-0} 

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3. Vu Trong Luong, Nguyen Thi Hue, “On the Asymptotic of Solution To the Dirichlet Problem For Hyperbolic Equations in Cylinders With Edges”, Electron. J. Qual. Theory Differ., 2014, no. 10, 1–15
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5. Mueller F. Schwab Ch., “Finite elements with mesh refinement for elastic wave propagation in polygons”, Math. Meth. Appl. Sci., 39:17 (2016), 5027–5042
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8. Korikov D. Plamenevskii B., “Asymptotics of Solutions to Nonstationary Maxwell System in Domains With Small Cavities”, 2018 Days on Diffraction (Dd), ed. Motygin O. Kiselev A. Goray L. Kazakov A. Kirpichnikova A. Perel M., IEEE, 2018, 176–181
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