Year: 2024
Type: preprint
Abstract: We investigate the inverse problem of recovering the diffusion and absorption coefficients $(σ,q)$ in the nonlocal diffuse optical tomography equation $(-\text{div}( σ\nabla))^s u+q u =0 \text{ in }Ω$ from the nonlocal Dirichlet-to-Neumann (DN) map $Λ^s_{σ,q}$. The purpose of this article is to establish the following approximation and uniqueness results. - Approximation: We show that solutions to the conductivity equation $ \text{div}( σ\nabla v)=0 \text{ in }Ω$ can be approximated in $H^1(Ω)$ by solutions to the nonlocal diffuse optical tomography equation and the DN map $Λ_σ$ related to conductivity equation can be approximated by the nonlocal DN map $Λ_{σ,q}^s$. - Local uniqueness: We prove that the absorption coefficient $q$ can be determined in a neighborhood $\mathcal{N}$ of the boundary $\partialΩ$ provided $σ$ is already known in $\mathcal{N}$. - Global uniqueness: Under the same assumptions as for the local uniqueness result, and if one of the potentials vanishes in $Ω$, then one can turn with the help of \ref{item 1 abstract} the local determination into a global uniqueness result. It is worth mentioning that the approximation result relies on the Caffarelli--Silvestre type extension technique and the geometric form of the Hahn--Banach theorem.
Source: arXiv (Cornell University)
Authors: Lin, Yi-Hsuan, Zimmermann, Philipp
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4399554171