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We study the inverse problem in Optical Tomography of determining the opticalproperties of a medium $\Omega\subset\mathbb{R}^n$, with $n\geq 3$, under theso-called diffusion approximation. We consider the time-harmonic case where$\Omega$ is probed with an input field that is modulated with a fixed harmonicfrequency $\omega=\frac{k}{c}$, where $c$ is the speed of light and $k$ is thewave number. Under suitable conditions that include a range of variability for$k$, we prove a result of H\"older stability of the derivatives of theabsorption coefficient $\mu_a$ of any order at the boundary $\partial\Omega$ interms of the measurements, in the case when the scattering coefficient $\mu_s$is assumed to be known. The stability estimates rely on the construction ofsingular solutions of the underlying forward elliptic system, which extendresults obtained in J. Differential Equations 84 (2): 252-272 for the singleelliptic equation.

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