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The asymptotic expansion of the trace of the heat kernel $\Theta(t) =\sum^\infty_{j=1}$ exp($-t\lambda_j)$ as $t\rightarrow 0^+$ has been derived for a variety of domains, where $\{\lambda_j\}$ are the eigenvalues of the negative Laplace operator $-\Delta = -\sum^2_{i=1}(\frac{\partial}{\partial x^i} )^2$ in the $(x^1, x^2)$-plane. The dependence of $\Theta(t)$ on the connectivity of domains and the boundary conditions is analyzed. Particular attention is given for a general annular drum in $\mathbb{R}^2$ together with Robin boundary conditions, where the coefficients in the boundary conditions are positive smooth functions. Some applications of an ideal gas enclosed in the general annular drum are given.

The object of this paper is to derive some interesting asymptotic formulae for spectra of a general convex domain in two or three dimensions, linked with variation of two positive distinct funcitons entering the boundary conditions on two “parts” of the smooth boundary of this domain. Further results may be obtained.