Monotonicity and local uniqueness for an isotropic nonlocal elliptic equation

Abstract

We extend monotonicity-based inversion methods to an inverse coefficient problem for the isotropic nonlocal elliptic equation \[ (-\nabla \cdot σ\nabla)^s u = 0 \quad \text{in } Ω\subset \mathbb{R}^n, \] where $0<s<1$, $n\ge 3$, 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.

arXiv comment17 pages. All