INVERSE PROBLEMS FOR A GENERAL MULTI-CONNECTED BOUNDED DRUM WITH APPLICATIONS IN PHYSICS

This paper studies the influence of a finite container on an ideal gas.The trace of the heat kernel （t） =exp, where are the eigenvalues of the negative Laplacian -in Rn（n = 2 or 3）, is studied for a general multi-connected bounded drum ft which is surrounded by simply connected bounded domains Ωi with smooth boundaries Ωi（i = 1,… ,m） where the Dirichlet, Neumann and Robin boundary conditions on Ωi（i = 1,…,m） are considered. Some geometrical properties of Ω are determined. The thermodynamic quantities for an ideal gas enclosed in Ω are examined by using the asymptotic expansions of （t） for short-time t. It is shown that the ideal gas can not feel the shape of its container Ω, although it can feel some geometrical properties of it.