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 except for a rectilinear moving slit, represented by a curve with Along 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

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