On a Calderón Problem in Frequency Differential Electrical Impedance Tomography

Recent research in electrical impedance tomography produced images of biological tissue from frequency differential boundary voltages and corresponding currents. Physically one is to recover the electrical conductivity $\sigma$ and permittivity $\epsilon$ from the frequency differential boundary data. Let $\gamma=\sigma+i\omega\epsilon$ denote the complex admittivity, $\Lambda_{\gamma}$ be the corresponding Dirichlet-to-Neumann map, and $\frac{d\Lambda_{\gamma}}{d\omega}|_{\omega=0}$ be its frequency differential at $\omega=0$. If $\sigma\in C^{1,1}(\overline\Omega)$ is constant near the boundary and $\epsilon\in C^{1,1}_0(\Omega)$, we show that $\frac{d\Lambda_{\gamma}}{d\omega}|_{\omega=0}$ uniquely determines $\nabla\cdot(\nabla\epsilon-\epsilon\nabla\ln\sigma)/\sigma$ inside $\Omega$. In addition, if $\Lambda_{\gamma}|_{\omega=0}$ is also known, then $\epsilon$ and $\sigma$ can be reconstructed inside. The method of proof uses the complex geometrical optics solutions.