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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the Eshelby-Kostrov property for the wave equation in the plane


Authors: M. A. Herrero, G. E. Oleaga and J. J. L. Velázquez
Journal: Trans. Amer. Math. Soc. 358 (2006), 3673-3695
MSC (2000): Primary 74R05, 74B05, 74G70
Published electronically: March 28, 2006
MathSciNet review: 2218994
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Abstract: This work deals with the linear wave equation considered in the whole plane $ \mathbb{R}^{2}$ except for a rectilinear moving slit, represented by a curve $ \Gamma\left( t\right) =\left\{ \left( x_{1},0\right) :-\infty<x_{1}<\lambda\left( t\right) \right\} $ with $ t\geq0.$ Along $ \Gamma\left( t\right) ,$ either homogeneous Dirichlet or Neumann boundary conditions are imposed. We discuss existence and uniqueness for these problems, and derive explicit representation formulae for solutions. The latter have a simple geometrical interpretation, and in particular allow us to derive precise asymptotic expansions for solutions near the tip of the curve. In the Neumann case, we thus recover a classical result in fracture dynamics, namely the form of the stress intensity factor in crack propagation under antiplane shear conditions.


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Additional Information

M. A. Herrero
Affiliation: Departamento de Matemática Aplicada, Facultad de Matemáticas, Universidad Complutense, Madrid 28040, Spain

G. E. Oleaga
Affiliation: Departamento de Matemática Aplicada, Facultad de Matemáticas, Universidad Complutense, Madrid 28040, Spain

J. J. L. Velázquez
Affiliation: Departamento de Matemática Aplicada, Facultad de Matemáticas, Universidad Complutense, Madrid 28040, Spain

DOI: http://dx.doi.org/10.1090/S0002-9947-06-03995-X
PII: S 0002-9947(06)03995-X
Received by editor(s): September 15, 2003
Received by editor(s) in revised form: September 12, 2004
Published electronically: March 28, 2006
Article copyright: © Copyright 2006 American Mathematical Society