Time-harmonic diffuse optical tomography: H\"older stability of the derivatives of the optical properties of a medium at the boundary

Abstract

We study the inverse problem in Optical Tomography of determining the optical properties of a medium $\Omega\subset\mathbb{R}^n$, with $n\geq 3$, under the so-called diffusion approximation. We consider the time-harmonic case where $\Omega$ is probed 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. Under suitable conditions that include a range of variability for $k$, we prove a result of H\"older stability of the derivatives of the absorption coefficient $\mu_a$ of any order at the boundary $\partial\Omega$ in terms of the measurements, in the case when the scattering coefficient $\mu_s$ is assumed to be known. The stability estimates 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.