Strong uniqueness principle for fractional polyharmonic operators and applications to inverse problems

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 $\tilde{P}((-{\Delta }_{\tilde{g}}{)}^{\tilde{s}})=0$ in a bounded Lipschitz domain $\Omega$ 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.

References in corpus (10)

• Simultaneous recoveries for semilinear parabolic systems
• On inverse problems in predator-prey models
• On inverse problems in multi-population aggregation models
• Determining a parabolic system by boundary observation of its non-negative solutions with biological applications
• Low regularity theory for the inverse fractional conductivity problem
• The fractional $p$-biharmonic systems: optimal Poincaré constants, unique continuation and inverse problems
• Determining Sources in the Bioluminescence Tomography Problem
• On the Obstacle Problem in Fractional Generalised Orlicz Spaces
• On an Anisotropic Fractional Stefan-Type Problem with Dirichlet Boundary Conditions
• Global Existence for Nonlocal Quasilinear Diffusion Systems in Non-Isotropic Non-Divergence Form