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

Herrero, M.; Oleaga, G.; Velázquez, J.

doi:10.1090/S0002-9947-06-03995-X

On the Eshelby-Kostrov property for the wave equation in the plane
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by M. A. Herrero, G. E. Oleaga and J. J. L. Velázquez PDF

Trans. Amer. Math. Soc. 358 (2006), 3673-3695 Request permission

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 $t\geq 0.$ 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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