Inverse problem for a general bounded domain in R-3 with piecewise smooth mixed boundary conditions

We study the influence of a finite container on an ideal gas. The trace of the heat kernel Theta(t) = Sigma(nu=1)(infinity) exp(-t mu(nu)), where \{mu(nu)\}(nu=1)(infinity) are the eigenvalues of the negative Laplacian -del(2) = -Sigma(beta=1)(3) (partial derivative/partial derivative x(beta))(2) in the (x(1), x(2), x(3))-space, is studied for a general bounded domain Omega with a smooth bounding surface S, where a finite number of Dirichlet, Neumann, and Robin boundary conditions on the piecewise smooth parts S-i (i = 1,..., n) of S are considered such that S = (Ui=1Si)-S-n. Some geometrical properties of Omega (the volume, the surface area, the mean curvature, and the Gaussian curvature) are determined. Furthermore, thermodynamic quantities, particularly the energy, for an ideal gas enclosed in the general bounded domain Omega with Dirichlet, Neumann, and Robin conditions are examined with the help of the asymptotic expansions of Theta(t) for short time t. We show that these thermodynamic quantities depend on some geometric properties of Omega.