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

Abstract

<p style='text-indent:20px;'>We address the inverse problem in Optical Tomography of stably determining the optical properties of an anisotropic medium <inline-formula><tex-math id="M3">\begin{document}$ \Omega\subset\mathbb{R}^n $\end{document}</tex-math></inline-formula>, with <inline-formula><tex-math id="M4">\begin{document}$ n\geq 3 $\end{document}</tex-math></inline-formula>, under the so-called <i>diffusion approximation</i>. Assuming that the <i>scattering coefficient</i> <inline-formula><tex-math id="M5">\begin{document}$ \mu_s $\end{document}</tex-math></inline-formula> is known, we prove H&#246;lder stability of the derivatives of any order of the <i>absorption coefficient</i> <inline-formula><tex-math id="M6">\begin{document}$ \mu_a $\end{document}</tex-math></inline-formula> at the boundary <inline-formula><tex-math id="M7">\begin{document}$ \partial\Omega $\end{document}</tex-math></inline-formula> in terms of the measurements, in the time-harmonic case, where the anisotropic medium <inline-formula><tex-math id="M8">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is interrogated with an input field that is modulated with a fixed harmonic frequency <inline-formula><tex-math id="M9">\begin{document}$ \omega = \frac{k}{c} $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M10">\begin{document}$ c $\end{document}</tex-math></inline-formula> is the speed of light and <inline-formula><tex-math id="M11">\begin{document}$ k $\end{document}</tex-math></inline-formula> is the wave number. The stability estimates are established under suitable conditions that include a range of variability for <inline-formula><tex-math id="M12">\begin{document}$ k $\end{document}</tex-math></inline-formula> and they rely on the construction of singular solutions of the underlying forward elliptic system, which extend results obtained in J. Differential Equations 84 (2): 252-272 for the single elliptic equation and those obtained in Applicable Analysis DOI:<a href="http://dx.doi.org/10.1080/00036811.2020.1758314" target="_blank">10.1080/00036811.2020.1758314</a>, where a Lipschitz type stability estimate of <inline-formula><tex-math id="M13">\begin{document}$ \mu_a $\end{document}</tex-math></inline-formula> on <inline-formula><tex-math id="M14">\begin{document}$ \partial\Omega $\end{document}</tex-math></inline-formula> was established in terms of the measurements.</p>