An inverse problem for a general annular-bounded domain in [formula omitted] with mixed boundary conditions and its physical applications

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 2 . The asymptotic expansion of the trace of the heat kernel θ( t)=∑ υ=1 ∞exp(− tμ ν ), where { μ ν } ν=1 ∞ are the eigenvalues of the negative Laplacian − Δ 2=−∑ k=1 2(∂/∂ x k ) 2 in the ( x 1, x 2)-plane, is studied for short-time t of a general annular-bounded domain Ω in R 2 together with its smooth inner boundary ∂Ω 1 and its smooth outer boundary ∂Ω 2 , where a finite number of Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth components Γ i ( i=1,…, m) of ∂Ω 1 and on the piecewise smooth components Γ i ( i= m+1,…, n) of ∂Ω 2 such that ∂Ω 1=⋃ i=1 mΓ i and ∂Ω 2=⋃ i=m+1 nΓ i , are considered. 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 Ω are given. Thermodynamic quantities of an ideal gas enclosed in Ω are examined. We use an asymptotic expansion for high temperatures to obtain the partition function of an ideal gas showing the leading corrections to the internal energy due to a finite container. We show that the ideal gas cannot feel the shape of its container, although it can feel some geometrical properties of it.