Monotonicity and local uniqueness for an isotropic nonlocal elliptic equation

TL;DR

The work develops a monotonicity framework for an isotropic nonlocal elliptic operator $(-\nabla\cdot\sigma\nabla)^s$ by relating coefficient orderings to the exterior Dirichlet-to-Neumann map $\Lambda_\sigma$ via the Caffarelli–Silvestre extension. It introduces Runge-type density results and constructs localized potentials for the extension problem, enabling a local uniqueness result: if $\Lambda_{\sigma_1}|_W=\Lambda_{\sigma_2}|_W$ under a sign condition on $\sigma_1-\sigma_2$ in a region $\mathcal{O}$, then $\sigma_1=\sigma_2$ in $\mathcal{O}$. The combination of monotonicity with localized potentials extends nonlocal inverse problem techniques to leading-coefficient identification and provides a pathway for local coefficient recovery from exterior data. These results augment global uniqueness by delivering a robust local identification mechanism for fractional elliptic problems.

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 $0 < s < 1$, $n \geq 3$, and $Ω$ is a bounded open set. We establish a monotonicity relation between the leading coefficient $σ$ 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.

Theorems & Definitions (17)

• Theorem 1.1: Local uniqueness
• Remark 1.2
• Lemma 3.1: Monotonicity relations
• Remark 3.2
• proof : Proof of Lemma \ref{['Lemma: monotonicity']}
• Corollary 3.3
• proof
• Remark 3.4
• Theorem 4.1: Runge approximation
• Corollary 4.2: Localized potentials