Asymptotic expansions of the heat kernel of the Laplacian for general annular bounded domains with Robin boundary conditions: Further results. (Q1421069)

The goal of this paper is to discuss the following general inverse problem: let \(\Omega\) be a general annular bounded domain in \(\mathbb{R}^N\) (\(N= 2\) or \(3\)) consisting of a simply connected bounded domain \(\Omega_1\) with a smooth boundary \(\partial\Omega_1\) and a simply connected bounded outer domain \(\Omega_1\supset\overline\Omega_1\) with a smooth boundary \(\partial\Omega_2\). Suppose that the eigenvalues are known exactly for the Helmholtz equation \[ \begin{cases} -\Delta_n\varphi= \lambda\varphi\quad &\text{in }\Omega,\\ ({\partial\over\partial n}+\gamma)\varphi= 0\quad &\text{on }\partial\Omega.\end{cases}\tag{1} \] The basic problem in this paper is to determine some geometric properties of the general annular bounded domain \(\Omega\) associated with (1) from the asymptotic expansion of \[ \theta(t):= \sum^\infty_{j=1} \exp(-t\lambda_j) \] as \(t\to+0\).