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

翻译：我们认为,通过最大限度地减少时间分解的Crank-Nicolson型功能,这是一个双曲自由边界问题。这一功能的特征是,在没有自由边界的情况下,它享有节能,这是进行数字计算的一个基本属性。显示最小化器的存在和规律性,并得出能源估计值。这些结果被用来表明一维环境中自由边界问题存在一个薄弱的解决方案。