The Calderón problem for a nonlocal diffusion equation with time-dependent coefficients

Abstract

We investigate the Calderón problem for a nonlocal diffusion equation depending on a globally unknown isotropic coefficient $\gamma (x,t)$. The forward problem is posed on $\Omega \times (0,T)$ for a domain $\Omega$ that is bounded in one direction. We first show that the Dirichlet-to-Neumann map ${\Lambda }_{\gamma }$ determines $\gamma$ in the measurement set. By studying various properties of the related nonlocal Neumann derivatives ${\mathcal{N}}_{\gamma }$, we prove that both quantities $⟨{\Lambda }_{\gamma }f,g⟩$ and $⟨{\mathcal{N}}_{\gamma }f,g⟩$ carry the same information as long as $f,g:{\mathbb{R}}^n\setminus \overline{\Omega }\to \mathbb{R}$ have disjoint supports and $\gamma$ is known in $\mathrm{supp}(f)\cup \mathrm{supp}(g)$. We obtain the desired global uniqueness theorem using a suitable integral identity for ${\mathcal{N}}_{\gamma }$ and the Runge approximation property. The results hold for any spatial dimension $n\ge 1$. In conclusion, the main observations of this article are twofold: (1) the information of ${\Lambda }_{\gamma }$ is needed for exterior determination for $\gamma$, (2) the knowledge of ${\mathcal{N}}_{\gamma }$ and $\gamma$ in the measurement set is enough to recover $\gamma$ in the interior.