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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

Elastodynamics in domains with edges


Authors: S. I. Matyukevich and B. A. Plamenevskii
Translated by: B. A. Plamenevskii
Original publication: Algebra i Analiz, tom 18 (2006), nomer 3.
Journal: St. Petersburg Math. J. 18 (2007), 459-510
MSC (2000): Primary 35L30, 35L35
Published electronically: April 11, 2007
MathSciNet review: 2255852
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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\geq 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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Additional Information

S. I. Matyukevich
Affiliation: St. Petersburg State University, Russia
Email: matsi@math.nw.ru

B. A. Plamenevskii
Affiliation: St. Petersburg State University, Russia
Email: plamen@rol.ru

DOI: http://dx.doi.org/10.1090/S1061-0022-07-00957-0
PII: S 1061-0022(07)00957-0
Keywords: Weighted estimates, asymptotics of solutions near edges, stress intensity factors
Received by editor(s): December 1, 2005
Published electronically: April 11, 2007
Additional Notes: Supported by RFBR (grant no. 05–01–01077)
Article copyright: © Copyright 2007 American Mathematical Society