On hearing the shape of a bounded domain with Robin boundary conditions

The asymptotic expansions of the trace of the heat kernel theta(t) = Sigma(j=1)(infinity) exp(-t lambda j) for small positive t, where \{lambda j\}(j=1)(infinity) are the eigenvalues of the negative Laplacian -Delta(n) = -Sigma(k=1)(n) (partial derivative/partial derivative x(k))(2) in R-n (n = 2 or 3), are studied for a general multiply connected bounded domain Omega which is surrounded by simply connected bounded domains Omega(i) with smooth boundaries partial derivative Omega(i)(i = 1,...,m), where smooth functions gamma(i)(i = 1, ..., m) are assuming the Robin boundary conditions (partial derivative/partial derivative n(i) + gamma(i))theta = 0 on partial derivative Omega(i). Here partial derivative/partial derivative n(i) denote differentiations dong the inward-pointing normals to partial derivative Omega(i)(i = 1, ...,m). Some applications of an ideal gas enclosed in the multiply connected bounded container with Neumann or Robin boundary conditions are given.