An inverse problem for a general vibrating annular membrane in R-3 with its physical applications: further results

This paper deals with the very interesting problem about the influence of the boundary conditions on the distribution of the eigenvalues of the negative Laplacian in R-3. The trace of the heat semigroup theta(t) = Sigma(v=1)(infinity) exp(-tmu(v)), where \{mu(v)\}(v=1)(infinity) are the eigenvalues of the negative Laplacian -del(2) = -Sigma(beta=1)(3) (partial derivative/partial derivativex(beta))(2) in the (x(1), x(2), x(3))-space, is studied for short-time t for a general vibrating annular membrane Omega in R-3 together with its smooth inner bounding surface S-1 and its smooth outer bounding surface S-2, where a finite number of Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth components S-i({*}) (i=1,...,m) of S-1 and on the piecewise smooth components S-i({*}) (i=m+1,..., n) of S-2 is considered such that S-1 = boolean ORi=1m S-i({*}) and S-2 = boolean ORi=m+1n S-i({*}). In this paper, one may extract information on the geometry of Omega by analyzing the asymptotic expansions of theta(t) for short-time t. Some applications of theta(t) for an ideal gas enclosed in the general vibrating annular membrane Omega are given. We show that the asymptotic expansion of theta(t) for short-time t plays an important role in investigating the influence of the annular region Omega on the thermodynamic quantities of an ideal gas. Some applications of theta(t) for an ideal gas enclosed in a compact n-dimensional Riemannian manifold are also given. We show that the ideal gas cannot feel the shape of its container, although it can feel some geometrical properties of it. (C) 2002 Elsevier Science Inc. All rights reserved.