An inverse problem for a general vibrating annular membrane in R3 with its physical applications: further results

摘要

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 R3. The trace of the heat semigroup θ(t)=∑v=1∞exp(−tμv), where {μv}∞v=1 are the eigenvalues of the negative Laplacian −∇2=−∑β=13∂/∂xβ2 in the (x1,x2,x3)-space, is studied for short-time t for a general vibrating annular membrane Ω in R3 together with its smooth inner bounding surface S1 and its smooth outer bounding surface S2, where a finite number of Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth components Si∗(i=1,…,m) of S1 and on the piecewise smooth components Si∗(i=m+1,…,n) of S2 is considered such that S1=⋃i=1mSi∗ and S2=⋃i=m+1nSi∗. 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.