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 Theta(t) = Sigma(mu=1)(infinity) exp(-tlambda(mu)), where \{lambda(mu)\}(mu=1)(infinity) are the eigenvalues of the negative Laplacian -Delta(n) = -Sigma(p=1)(n) (partial derivative/partial derivativex(p))(2) in R-n (n = 2 or 3), is studied for a general multinegative connected bounded drum Omega which is surrounded by simply connected bounded domains Omega(i) with smooth boundaries partial derivativeOmega(i)(i = 1,...,m) where the Dirichlet, Neumann and Robin boundary conditions on partial derivativeOmega(i)(i = 1,...,m) are considered. Some geometrical properties of Omega are determined. The thermodynamic quantities for an ideal gas enclosed in Omega axe examined by using the asymptotic expansions of Theta(t) for short-time t. It is shown that the ideal gas can not feel the shape of its container Omega, although it can feel some geometrical properties of it.