An inverse problem for a general annular drum with positive smooth functions in the Robin boundary conditions

Abstract

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