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 $$(-\nabla \cdot \sigma \nabla)^s u = 0 \quad \text{in } \Omega \subset \mathbb{R}^n,$$ where $0 < s < 1$, $n \geq 3$, and $\Omega$ 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.