Asymptotic expansions of the heat kernel of the Laplacian for general annular bounded domains with Robin boundary conditions: Further results

The asymptotic expansions of the trace of the heat kernel Theta(t) = Sigma(v=1)(infinity) exp(-tlambda(v)) for small positive t, where \{lambdav\} are the eigenvalues of the negative Laplacian -Delta(n) = -Sigma(i=1)(n)((partial derivative)/(partial derivativexi))(2) in R-n (n = 2 or 3), are studied for a general annular bounded domain Omega with a smooth inner boundary partial derivativeOmega(1) and a smooth outer boundary partial derivativeOmega(2), where a finite number of piecewise smooth Robin boundary conditions ((partial derivative)/(partial derivativenj) + gamma(j))phi = 0 on the components Gamma(j) (j = 1, ...,k) of partial derivativeOmega(1) and on the components Gamma(j) (j = k + 1, ...,m) of partial derivativeOmega(2) are considered such that partial derivativeOmega(1) = boolean OR(j=1)(k)Gamma(j) and partial derivativeOmega(2) = boolean OR(j=k+1)(m)Gamma(j) and where the coefficients gamma(j)(j = 1,...,m) are piecewise smooth positive functions. Some applications of Theta(t) for an ideal gas enclosed in the general annular bounded domain Omega are given. Further results are also obtained.