An Inverse Problem for a General Doubly-Connected Bounded Domain with Impedance Boundary Conditions

E.M.E. Zayed

Abstract: The spectral function $\theta(t)=\sum_{\nu=1}^{\infty}\exp(-t\,\lambda_{\nu})$, where $\{\lambda_{\nu}\}_{\nu=1}^{\infty}$ are the eigenvalues of the negative Laplacian $-\Delta=-\sum_{i=1}^{2}(\frac{\partial}{\partial x^{i}})^{2}$ in the $(x^{1},x^{2})$-plane, is studied for a general doubly-connected bounded domain $\Omega$ in $\R^{2}$ together with its smooth inner boundary $\partial\Omega_{1}$ and its smooth outer boundary $\partial\Omega_{2}$, where piecewise smooth impedance boundary conditions on the two parts $\Gamma_{1}$, $\Gamma_{2}$ of $\partial\Omega_{1}$ and on the two parts $\Gamma_{3}$, $\Gamma_{4}$ of $\partial\Omega_{2}$ are considered, such that $\partial\Omega_{1}=\Gamma_{1}\cup\Gamma_{2}$ and $\partial\Omega_{2}=\Gamma_{3}\cup\Gamma_{4}$.

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