Entanglement principle and fractional Calder\'on problem for nonlocal parabolic operators

TL;DR

This paper introduces a new entanglement principle for nonlocal parabolic operators and demonstrates its use in uniquely determining lower-order perturbations from boundary data, advancing inverse problem solutions.

Contribution

It presents a novel entanglement principle for variable-coefficient nonlocal parabolic operators and adapts it for inverse problems with limited solution regularity.

Findings

Established a new entanglement principle for nonlocal parabolic operators.

Proved unique determination of lower-order perturbations from Dirichlet-to-Neumann map.

Developed a modified entanglement principle to handle regularity issues.

Abstract

We examine inverse problems for the variable-coefficient nonlocal parabolic operator $(\partial_t - \Delta_g)^s$, where $0 < s < 1$. This article makes two primary contributions. First, we introduce a novel entanglement principle for these operators under suitable smoothness conditions. Second, we prove that lower-order perturbations can be uniquely determined from the associated Dirichlet-to-Neumann map using this principle. However, due to insufficient solution regularity, direct application of the entanglement principle to the inverse problem is not feasible. To address this, we derive a modified entanglement principle, enabling the effective resolution of related inverse problems.