Time-harmonic diffuse optical tomography: Holder stability of the derivatives of the optical properties of a medium at the boundary: 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 Ω ⊂ Rⁿ, with n ≥ 3, under the so-called diffusion approximation. Assuming that the scattering coefficient µ_s is known, we prove Hölder stability of the derivatives of any order of the absorption coefficient µ_a at the boundary ∂Ω in terms of the measurements, in the time-harmonic case, where the anisotropic medium Ω is interrogated with an input field that is modulated with a fixed harmonic frequency ω =^k, where c is the speed of light and k is the wave number. c 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 µ_a on ∂Ω was established in terms of the measurements.