On hearing the shape of the three-dimensional multi-connected vibrating membrane with piecewise smooth boundary conditions

Abstract

This paper deals with the very interesting problem about the influence of a finite number of piecewise smooth mixed boundary conditions on the distribution of the eigenvalues of the negative Laplacian in R3. The trace of the heat kernel are the eigenvalues of the negative Laplacian -space, is studied for a general multiplyconnected bounded domain ω in R3 surrounding by simply connected bounded domains ωj with smooth bounding surfaces Sj(j = I ,…,n), where a finite number of Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth components of the bounding surfaces Sj is considered, such that where k0 = 0. Some applications of ø(t) for an ideal gas enclosed in the multiply-connected bounded container ω with Dirichlet, Neumann and Robin boundary conditions are given. We show that the asymptotic expansion of ø(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. We also show that the ideal gas can not feel the shape of its container, although it can feel some geometrical properties of it.