A Calderón problem with frequency-differential data in dispersive media

Abstract:

We consider the problem of identifying a complex valued coefficient $\gamma (x,\omega )$ in the conductivity equation $\nabla \cdot \gamma (\cdot ,\omega )\nabla u(\cdot ,\omega )=0$ from knowledge of the frequency differentials of the Dirichlet-to-Neumann map. For a frequency analytic $\gamma (\cdot ,\omega )=\sum _{k=0}^\infty (\sigma _k+i\epsilon _k)\omega ^k$, in three dimensions and higher, we show that $\left .\frac {d^j}{d\omega ^j}\Lambda _{\gamma (\cdot ,\omega )}\right |_{\omega =0}$ for $j=0,1,\dots ,N$ recovers $\sigma _0,\dots , \sigma _N$ and $\epsilon _1,\dots ,\epsilon _N$. This problem arises in frequency differential electrical impedance tomography of dispersive media.