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

This paper deals with the very interesting problem concerning the influence of the boundary conditions on the distribution of the eigenvalues of the negative Laplacian in R-3. The trace of the heat semigroup theta(t) = Sigma(upsilon=1)(infinity) exp(-tmu(upsilon)), where \{mu(upsilon)\}(upsilon=1)(infinity) are the eigenvalues of the negative Laplacian -del(2) = E-beta=1(3)(partial derivative/partial derivativex(beta))(2) in the (x(1), x(2), x(3))-space, is studied for a general multiply-connected bounded domain Q in R' surrounding by simply connected bounded domains Omega(j) with smooth bounding surfaces S-j (j = 1,..., n), where a finite number of piecewise smooth Robin boundary conditions on the piecewise smooth components S-i{*} (i = 1 + k(j- 1),. . . k(j)) of the bounding surfaces S-j is considered, such that kj S-j = U-i=1+kj-1(kj) S-i{*}, where k(0) = 0. Some applications of theta(t) for an ideal gas enclosed in the multiply-connected bounded container Q with Robin boundary conditions are given. We show that the asymptotic expansion of theta(t) for short-time t plays an important role in investigating the influence of the finite container Omega on the thermodynamic quantities of an ideal gas.