Time-harmonic diffuse optical tomography: Hölder stability of the derivatives of the optical properties of a medium at the boundary

Abstract

We address the inverse problem in Optical Tomography of stably determining the optical properties of an anisotropic medium \begin{document}$ \Omega\subset\mathbb{R}^n $\end{document}, with \begin{document}$ n\geq 3 $\end{document}, under the so-called diffusion approximation. Assuming that the scattering coefficient \begin{document}$ \mu_s $\end{document} is known, we prove Hölder stability of the derivatives of any order of the absorption coefficient \begin{document}$ \mu_a $\end{document} at the boundary \begin{document}$ \partial\Omega $\end{document} in terms of the measurements, in the time-harmonic case, where the anisotropic medium \begin{document}$ \Omega $\end{document} is interrogated with an input field that is modulated with a fixed harmonic frequency \begin{document}$ \omega = \frac{k}{c} $\end{document}, where \begin{document}$ c $\end{document} is the speed of light and \begin{document}$ k $\end{document} is the wave number. The stability estimates are established under suitable conditions that include a range of variability for \begin{document}$ k $\end{document} and they rely on the construction of singular solutions of the underlying forward elliptic system, which extend results obtained in J. Differential Equations 84 (2): 252-272 for the single elliptic equation and those obtained in Applicable Analysis DOI:10.1080/00036811.2020.1758314, where a Lipschitz type stability estimate of \begin{document}$ \mu_a $\end{document} on \begin{document}$ \partial\Omega $\end{document} was established in terms of the measurements.