Inverse problems for a general multi-connected bounded drum with applications in physics

In this paper, we study 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 multi-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 are examined by using the asymptotic expansions of Theta(t) for short-time t. We show that the ideal gas cannot feel the shape of its container Omega, although it can feel some geometrical properties of it. (C) 2002 Elsevier Science Inc. All rights reserved.