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

摘要

In this paper, we study the influence of a finite container on an ideal gas. The trace of the heat kernel Θ(t)=∑μ=1∞exp(−tλμ), where {λμ}μ=1∞ are the eigenvalues of the negative Laplacian −Δn=−∑p=1n∂/∂xp2 in Rn (n=2 or 3), is studied for a general multi-connected bounded drum Ω which is surrounded by simply connected bounded domains Ωi with smooth boundaries ∂Ωi(i=1,…,m) where the Dirichlet, Neumann and Robin boundary conditions on ∂Ωi(i=1,…,m) are considered. Some geometrical properties of Ω are determined. The thermodynamic quantities for an ideal gas enclosed in Ω are examined by using the asymptotic expansions of Θ(t) for short-time t. We show that the ideal gas cannot feel the shape of its container Ω, although it can feel some geometrical properties of it.