Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems

Abstract

We prove a unique continuation property for the fractional Laplacian $(-\Delta {)}^s$ when $s\in (-n/2,\infty )\setminus \mathbb{Z}$. In addition, we study Poincaré-type inequalities for the operator $(-\Delta {)}^s$ when $s\ge 0$. We apply the results to show that one can uniquely recover, up to a gauge, electric and magnetic potentials from the Dirichlet-to-Neumann map associated to the higher order fractional magnetic Schrödinger equation. We also study the higher order fractional Schrödinger equation with singular electric potential. In both cases, we obtain a Runge approximation property for the equation. Furthermore, we prove a uniqueness result for a partial data problem of the $d$-plane Radon transform in low regularity. Our work extends some recent results in inverse problems for more general operators.

arXiv comment35 pages, final version

References in corpus (22)

• An extension problem related to the fractional Laplacian
• Fractional Quantum Mechanics and Levy Path Integrals
• Fractional Quantum Mechanics
• Ten equivalent definitions of the fractional Laplace operator
• Nonlocal diffusion and applications
• Uniqueness and reconstruction for the fractional Calderón problem with a single measurement
• The Calderón problem for the fractional Schrödinger equation
• The fractional Calderón problem: low regularity and stability
• Unique Continuation for Fractional Schrödinger Equations with Rough Potential
• On higher order extensions for the fractional Laplacian
• Sobolev spaces on non-Lipschitz subsets of ${\mathbb{R}}^n$ with application to boundary integral equations on fractal screens
• Monotonicity-based inversion of the fractional Schrödinger equation I. Positive potentials
• Exponential instability in the fractional Calderón problem
• Monotonicity-based inversion of the fractional Schrödinger equation II. General potentials and stability
• Multiplication in Sobolev Spaces, Revisited
• The Calderón Problem For The Fractional Magnetic Operator
• On an inverse problem for a fractional semilinear elliptic equation involving a magnetic potential
• Unique continuation of the normal operator of the X-ray transform and applications in geophysics
• On Radon transforms on tori
• An iterative inversion of weighted Radon transforms along hyperplanes
• Fourier analysis of periodic Radon transforms
• An example of non-uniqueness for the weighted Radon transforms along hyperplanes in multidimensions

Cited by in corpus (7)

• The higher order fractional Calderón problem for linear local operators: uniqueness
• Low regularity theory for the inverse fractional conductivity problem
• The fractional $p$-biharmonic systems: optimal Poincaré constants, unique continuation and inverse problems
• An inverse problem for a fractional diffusion equation with fractional power type nonlinearities
• Stability estimates for the inverse fractional conductivity problem
• Inverse problems for the fractional Laplace equation with lower order nonlinear perturbations
• Partial data problems in scalar and vector field tomography