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

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 for small t and , where {μν}ν=1∞ are the eigenvalues of the negative Laplacian in the (x1,x2)-plane, is studied for an annular vibrating membrane Ω in R2 together with its smooth inner boundary ∂Ω1 and its smooth outer boundary ∂Ω2, where a finite number of Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth components Γj(j=1,…,m) of ∂Ω1 and on the piecewise smooth components Γj (j=m+1,…,n) of ∂Ω2 such that ∂Ω1= j=1mΓj and ∂Ω2= j=m+1nΓj 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 for small t. Some applications of for an ideal gas enclosed in the general annular bounded domain Ω are given.