Short-time asymptotics of the two-dimensional wave equation for an annular vibrating membrane with applications in the mathematical physics

✍ Scribed by E.M.E. Zayed

• Publisher: Elsevier Science
• Year: 2004
• Tongue: English
• Weight: 152 KB
• Volume: 19
• Category: Article
• ISSN: 0960-0779
• DOI: 10.1016/s0960-0779(03)00200-5

✦ Synopsis

We study the influence of a finite container on an ideal gas using the wave equation approach. The asymptotic expansion of the trace of the wave kernel l lðtÞ

, where fl m g 1 m¼1 are the eigenvalues of the negative Laplacian ÀD ¼ À

, is studied for an annular vibrating membrane X in R 2 together with its smooth inner boundary oX 1 and its smooth outer boundary oX 2 , where a finite number of Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth components C j ðj ¼ 1; . . . ; mÞ of oX 1 and on the piecewise smooth components

The basic problem is to extract information on the geometry of the annular vibrating membrane X from complete knowledge of its eigenvalues using the wave equation approach by analyzing the asymptotic expansions of the spectral function l lðtÞ for small jtj. Some applications of l lðtÞ for an ideal gas enclosed in the general annular bounded domain X are given.