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

Abstract

This work deals with the linear wave equation considered in the whole plane R2 except for a rectilinear moving slit, represented by a curve Γ (t) = {(x1, 0) : −∞ < x1 < λ(t)} with t ≥ 0. Along Γ (t) , either homogeneous Dirichlet or Neumann boundary conditions are imposed. We discuss existence and uniqueness for these problems, and derive explicit representation formulae for solutions. These last have a simple geometrical interpretation, and in particular allow 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