An inverse problem for a general vibrating annular membrane in [formula omitted] with its physical applications: further results

Abstract

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 θ( t)=∑ v=1 ∞exp(− tμ v ), where { μ v } ∞ v=1 are the eigenvalues of the negative Laplacian −∇ 2=−∑ β=1 3 ∂/ ∂x β 2 in the ( x 1, x 2, x 3)-space, is studied for short-time t for a general vibrating annular membrane Ω 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=⋃ i=1 mS i ∗ and S 2=⋃ i=m+1 nS i ∗ . In this paper, one may extract information on the geometry of Ω by analyzing the asymptotic expansions of θ( t) for short-time t. Some applications of θ( t) for an ideal gas enclosed in the general vibrating annular membrane Ω are given. We show that the asymptotic expansion of θ( t) for short-time t plays an important role in investigating the influence of the annular region Ω on the thermodynamic quantities of an ideal gas. Some applications of θ( 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.