Strong uniqueness principle for fractional polyharmonic operators and applications to inverse problems
Ching-Lung Lin, Hongyu Liu, Catharine W. K. Lo
TL;DR
This work develops a theoretical framework for poly-fractional operators $P((-\Delta_g)^s)=\sum_{i=1}^M \alpha_i(-\Delta_{g_i})^{s_i}$ with mixed fractional orders and anisotropies, formulating an exterior-value problem and a Dirichlet-to-Neumann map for identifiability. A novel unique continuation principle tailored to space-domain poly-fractional operators underpins the main results, enabling uniqueness in recovering the potential $q$, semilinear source $F$, and non-isotropy coefficients from exterior measurements (sometimes with a single measurement, and via higher-order linearisation for the semilinear case). The contributions extend the fractional Calderón paradigm to mixed-order nonlocal operators, addressing both nonlocal and local interactions and providing a path toward practical diffusion-property identification in complex media. The results have potential applications in diffusion modeling, tomography, and imaging where multiple scales of nonlocal and local processes interact.
Abstract
In this work, we are concerned with inverse problems involving poly-fractional operators, where the poly-fractional operator is of the form \[P( (-Δ_g)^s)u := \sum_{i=1}^M α_i(-Δ_{g_i})^{s_i}u\] for $s=(s_1,\dots,s_M)$, $0<s_1<\cdots<s_M<\infty$, $s_M\in\mathbb{R}_+\backslash\mathbb{Z}$, $g=(g_1,\dots,g_M)$. There are three major contributions in this work that are new to the literature. First, we propose equations involving such poly-fractional operators $P$, which have not been previously considered in the general setting. Such equations arise naturally from the superposition of multiple stochastic processes with different scales, including classical random walks and Lévy flights. Secondly, we give novel results for the unique continuation properties for fractional polyharmonic $u$, in the sense that $u$ satisfies $\tilde{P}((-Δ_{\tilde{g}})^{\tilde{s}})=0$ in a bounded Lipschitz domain $Ω$ for some $\tilde{P}$. With these results in hand, we consider the inverse problems for $P$, and proved the uniqueness in recovering the potential, the source function in the semilinear case, and the coefficients associated to the non-isotropy of the fractional operator.