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

We address the inverse problem in Optical Tomography of stably determining the optical properties of an anisotropic medium $ \Omega\subset\mathbb{R}^n $, with $ n\geq 3 $, under the so-called diffusion approximation. Assuming that the scattering coefficient $ \mu_s $ is known, we prove Hölder stability of the derivatives of any order of the absorption coefficient $ \mu_a $ at the boundary $ \partial\Omega $ in terms of the measurements, in the time-harmonic case, where the anisotropic medium $ \Omega $ is interrogated with an input field that is modulated with a fixed harmonic frequency $ \omega = \frac{k}{c} $, where $ c $ is the speed of light and $ k $ is the wave number. The stability estimates are established under suitable conditions that include a range of variability for $ k $ 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 $ \mu_a $ on $ \partial\Omega $ was established in terms of the measurements. [ABSTRACT FROM AUTHOR]