An analysis of the nonlinear equation of motion of a vibrating membrane in the space of BV functions

Abstract

In this article the nonlinear equation of motion of vibrating membrane $u_{tt}-div\left\{{\sqrt{1+|Vu{|}^2}}^{-1}Vu\right\}=0$ is discussed in the space of functions having bounded variation. Approximate solutions are constructed in Rothe's method. It is proved that a subsequence of them converges to a function $u$ and that, if $u$ satisfies the energy conservation law, then it is a weak solution in the space of functions having bounded variation. The main tool is varifold convergence.