On hearing the shape of a general multi-connected vibrating membrane in R-2 with piecewise smooth positive functions in the Robin boundary conditions

The asymptotic expansion of the trace of the heat kernel Theta(t) =Sigma(J=1)(infinity) exp(-tlambda(J)) as t -> O+ has been derived for a variety of domains, where \{lambda(J)\}\} are the eigenvalues of the negative Laplace operator {[}GRAPHICS] in the (x(1),x(2))-plane. The dependence of Theta(t) on the connectivity of domains and the boundary conditions are analyzed. Particular attention is given for a general multiply connected bounded domain in R-2 together with a finite number of piecewise Robin boundary conditions, where the coefficients in the boundary conditions are piecewise smooth positive functions. Some applications of an ideal gas enclosed in the general multiply connected domain are given. (C) 2002 Elsevier Science Inc. All rights reserved.