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Entanglement principle and fractional Calderón problem for nonlocal parabolic operators

Ru-Yu Lai, Yi-Hsuan Lin, Lili Yan

TL;DR

This work addresses inverse problems for the nonlocal parabolic operator $\mathcal{H}_g^s=(\partial_t-\Delta_g)^s$ with $0<s<1$ by introducing an entanglement principle that decouples mixed fractional effects under a nonresonance condition on exponents. A modified entanglement principle is developed to overcome solution regularity barriers, enabling the analysis of poly-parabolic models $P_V=\sum_{k=1}^N b_k \mathcal{H}_g^{s_k}+V$ and the recovery of the time-dependent potential $V$ from exterior Dirichlet-to-Neumann data. The paper proves (i) an entanglement principle for fractional parabolic operators with $\alpha_k$ nonresonant, (ii) global uniqueness for $V$ from the DN map, and (iii) a Runge approximation framework to connect exterior data with interior information. The results extend nonlocal Calderón-type theory to time-dependent multi-term fractional parabolic models, providing a rigorous pathway for unique recovery of lower-order perturbations in practical, variable-coefficient settings through exterior measurements.

Abstract

We examine inverse problems for the variable-coefficient nonlocal parabolic operator $(\partial_t - Δ_g)^s$, 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.

Entanglement principle and fractional Calderón problem for nonlocal parabolic operators

TL;DR

This work addresses inverse problems for the nonlocal parabolic operator with by introducing an entanglement principle that decouples mixed fractional effects under a nonresonance condition on exponents. A modified entanglement principle is developed to overcome solution regularity barriers, enabling the analysis of poly-parabolic models and the recovery of the time-dependent potential from exterior Dirichlet-to-Neumann data. The paper proves (i) an entanglement principle for fractional parabolic operators with nonresonant, (ii) global uniqueness for from the DN map, and (iii) a Runge approximation framework to connect exterior data with interior information. The results extend nonlocal Calderón-type theory to time-dependent multi-term fractional parabolic models, providing a rigorous pathway for unique recovery of lower-order perturbations in practical, variable-coefficient settings through exterior measurements.

Abstract

We examine inverse problems for the variable-coefficient nonlocal parabolic operator , where . 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.
Paper Structure (13 sections, 9 theorems, 141 equations)

This paper contains 13 sections, 9 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Mathematical formulations and main results
  3. Previous work on nonlocal parabolic inverse problems
  4. Main results
  5. Organization of the article.
  6. Preliminaries
  7. Function spaces
  8. The nonlocal parabolic operator
  9. The well-posedness
  10. The Dirichlet-to-Neumann map
  11. The entanglement principle
  12. Inverse problems and proof of main results
  13. Global uniqueness for the fractional poly-parabolic operators

Key Result

Theorem 1.2

Let $\mathcal{O}\subset {\mathbb R}^n$ be a nonempty open set for $n\geq 2$. Let $N\in\mathbb{N}$, $T>0$, and $\{\alpha_k\}_{k=1}^N \subset (0,\infty)\setminus {\mathbb N}$ satisfy Assumption exponent condition. Suppose $g \in C^\infty({\mathbb R}^n;{\mathbb R}^{n\times n})$ satisfy ellipticity. Ass for $k=1,\ldots, N$, where $D_{x,t}^\beta = \frac{\partial^{|\beta|}}{\partial_t^{\beta_0}\partial

Theorems & Definitions (27)

  • Remark 1.1
  • Theorem 1.2: Entanglement principle
  • Remark 1.3
  • Theorem 1.4: Global uniqueness
  • Definition 2.1: Balakrishnan formula
  • Remark 2.2
  • Definition 2.3
  • Theorem 2.4: Well-posedness
  • proof
  • Remark 2.5
  • ...and 17 more