Existence and Uniqueness of a Solution in the Space of BV Functions to the Equation of a Vibrating Membrane with a “Viscosity” Term

Abstract

A nonlinear equation of motion of vibrating membrane with a "viscosity" term is investigated. Usually, the term −Δ is added, and it is well known that this equation is well posed in the space of 1,2 functions. In this paper, the viscosity term is changed to −(div(∇ /√1 + |∇ | 2 )) , and it is proved that if initial data is slightly smooth (but belonging to 2,2 is sufficient), then a weak solution exists uniquely in the space of BV functions.