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

We study the inverse problem in Optical Tomography of determining the optical properties of a medium $\mathrm{\Omega }\subset {\mathbb{R}}^n$, with $n\ge 3$, under the so-called diffusion approximation. We consider the time-harmonic case where $\mathrm{\Omega }$ 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 $\mathrm{\partial }\mathrm{\Omega }$ 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.