A remark on Dirichlet boundary condition for the nonlinear equation of motion of a vibrating membrane

✍ Scribed by K. Kikuchi

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 469 KB
• Volume: 47
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/s0362-546x(01)00244-9

✦ Synopsis

The author treated in his previous work [4] the nonlinear equation of motion of vibrating membrane (u_{t t}-\operatorname{div}\left{\left(1+|\nabla u|^{2}\right)^{-1 / 2} \nabla u\right}=0) in the space of functions having bounded variation and constructed approximate solutions in Rothe's method. He has established 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. In [4] Dirichlet condition is defined as (\gamma u=0). In this article Dirichlet condition is treated in a weaker sense than (\gamma u=0). This is possibly more natural in treating this equation in the space of functions having bounded variation. The same fact as in [4] is established under this weaker condition.