An inverse problem for the three-dimensional multi-connected vibrating membrane with Robin boundary conditions

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)=∑∞ν=1exp(−tμν), where {μν}ν=1∞ are the eigenvalues of the negative Laplacian −∇2=−∑3β=1(∂/∂xβ)2 in the (x1,x2,x3)-space, is studied for a general multiply connected bounded domain Ω in R3 surrounded by simply connected bounded domains Ωj with smooth bounding surfaces Sj (j=1,…,n), where a finite number of piecewise smooth Robin boundary conditions on the piecewise smooth components Si∗ (i=1+kj−1,…,kj) of the bounding surfaces Sj are considered, such that Sj=⋃kji=1+kj−1Si∗, where k0=0. Some applications of θ(t) for an ideal gas enclosed in the multiply connected bounded container Ω with Robin boundary conditions 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 finite container Ω on the thermodynamic quantities of an ideal gas.