A Crank-Nicolson type minimization scheme for a hyperbolic free boundary problem

<p style='text-indent:20px;'>We consider a hyperbolic free boundary problem by means of minimizing time discretized functionals Crank-Nicolson type. The feature this functional is that it enjoys energy conservation in the absence boundaries, which an essential property for numerical calculations. existence and regularity minimizers shown estimate derived. These results are then used to show weak solution 1-dimensional setting.</p>