Monotonicity and local uniqueness for an isotropic nonlocal elliptic equation

We extend monotonicity-based inversion methods to an inverse coefficient problem for the isotropic nonlocal elliptic equation $$(-\mathrm{\nabla }\cdot \sigma \mathrm{\nabla }{)}^{s}u=0\text{in }\mathrm{\Omega }\subset {\mathbb{R}}^n,$$ where $0<s<1$, $n\ge 3$, and $\mathrm{\Omega }$ is a bounded open set. We establish a monotonicity relation between the leading coefficient $\sigma$ and the (partial) exterior Dirichlet-to-Neumann (DN) map. Our main result shows that a monotonicity ordering of the coefficients implies a corresponding ordering of the DN maps. Furthermore, we construct localized potentials for the nonlocal equation, which yield a local uniqueness result for the fractional inverse problem.