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

This article is cited in 8 scientific papers (total in 8 papers)

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
Received: 01.12.2005

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
\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
\jour St. Petersburg Math. J.
\yr 2007
\vol 18
\issue 3
\pages 459--510

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    This publication is cited in the following articles:
    1. Hung N.M., Anh C.T., “Asymptotic Expansions of Solutions of the First Initial Boundary-Value Problem for Schrodinger Systems in Domains with Conical Points. II”, Ukrainian Mathematical Journal, 61:12 (2009), 1923–1945  crossref  mathscinet  zmath  isi  scopus
    2. D. V. Korikov, “Asymptotic behavior of solutions to wave equation in domain with a small hole”, St. Petersburg Math. J., 26:5 (2015), 813–838  mathnet  crossref  mathscinet  isi  elib  elib
    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  crossref  mathscinet  isi
    4. D. V. Korikov, B. A. Plamenevskiǐ, “Asymptotics of solutions for stationary and nonstationary Maxwell systems in a domain with small holes”, St. Petersburg Math. J., 28:4 (2017), 507–554  mathnet  crossref  mathscinet  isi  elib
    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  crossref  mathscinet  zmath  isi  scopus
    6. Gimperlein H. Meyer F. Oezdemir C. Stark D. Stephan E.P., “Boundary Elements With Mesh Refinements For the Wave Equation”, Numer. Math., 139:4 (2018), 867–912  crossref  mathscinet  zmath  isi  scopus
    7. Mueller F. Schotzau D. Schwab Ch., “Discontinuous Galerkin Methods For Acoustic Wave Propagation in Polygons”, J. Sci. Comput., 77:3, SI (2018), 1909–1935  crossref  mathscinet  zmath  isi  scopus
    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  crossref  isi
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