Abstract

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 μ ̂ (t)=∑ υ=1 ∞ exp(− itμ υ 1/2) for small | t| and i= −1 , where { μ ν } ν=1 ∞ are the eigenvalues of the negative Laplacian −Δ=−∑ k=1 2( ∂ ∂x k ) 2 in the ( x 1, x 2)-plane, is studied for an annular vibrating membrane Ω in R 2 together with its smooth inner boundary ∂Ω 1 and its smooth outer boundary ∂Ω 2 , where a finite number of Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth components Γ j ( j=1,…, m) of ∂Ω 1 and on the piecewise smooth components Γ j (j=m+1,…,n) of ∂Ω 2 such that ∂Ω 1=⋃ j=1 mΓ j and ∂Ω 2=⋃ j=m+1 nΓ j is considered. The basic problem is to extract information on the geometry of the annular vibrating membrane Ω from complete knowledge of its eigenvalues using the wave equation approach by analyzing the asymptotic expansions of the spectral function μ ̂ (t) for small | t|. Some applications of μ ̂ (t) for an ideal gas enclosed in the general annular bounded domain Ω are given.

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