## Journal of the Mathematical Society of Japan

- J. Math. Soc. Japan
- Volume 52, Number 4 (2000), 741-766.

### 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.

#### Article information

**Source**

J. Math. Soc. Japan, Volume 52, Number 4 (2000), 741-766.

**Dates**

First available in Project Euclid: 10 June 2008

**Permanent link to this document**

https://projecteuclid.org/euclid.jmsj/1213107108

**Digital Object Identifier**

doi:10.2969/jmsj/05240741

**Mathematical Reviews number (MathSciNet)**

MR1774628

**Zentralblatt MATH identifier**

0964.35101

**Subjects**

Primary: 35L70: Nonlinear second-order hyperbolic equations 49J40: Variational methods including variational inequalities [See also 47J20] 49Q15: Geometric measure and integration theory, integral and normal currents [See also 28A75, 32C30, 58A25, 58C35]

**Keywords**

Hyperbolic equations BV functions Rothe's method direct variational method varifolds

#### Citation

KIKUCHI, Koji. An analysis of the nonlinear equation of motion of a vibrating membrane in the space of BV functions. J. Math. Soc. Japan 52 (2000), no. 4, 741--766. doi:10.2969/jmsj/05240741. https://projecteuclid.org/euclid.jmsj/1213107108