Inverse problem for a nonlocal diffuse optical tomography equation

Abstract

In this article a nonlocal analogue of an inverse problem in diffuse optical tomography is considered. We show that whenever one has given two pairs of diffusion and absorption coefficients $(\gamma_j,q_j)$, $j=1,2$, such that there holds $q_1=q_2$ in the measurement set $W$ and they generate the same DN data, then they are necessarily equal in $\mathbb{R}^n$ and $\Omega$, respectively. Additionally, we show that the condition $q_1|_W=q_2|_W$ is optimal in the sense that without this restriction one can construct two distinct pairs $(\gamma_j,q_j)$, $j=1,2$ generating the same DN data.