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

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.