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

Giovanni Covi; Keijo Mönkkönen; Jesse Railo

doi:10.3934/ipi.2021009

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2021, Volume 15, Issue 4: 641-681. Doi: 10.3934/ipi.2021009

This issue Previous Article Cauchy problem of non-homogenous stochastic heat equation and application to inverse random source problem Next Article Bragg scattering tomography

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

1.
Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD) FI-40014 University of Jyväskylä, Finland
2.
Seminar for Applied Mathematics, Department of Mathematics, ETH Zurich, Rämistrasse 101, CH-8092 Zürich, Switzerland

^* Corresponding author
^* Corresponding author

Received: March 2020

Revised: September 2020

Early access: January 2021

Published: August 2021

G.C. was partially supported by the European Research Council under Horizon 2020 (ERC CoG 770924). K.M. and J.R. were partially supported by Academy of Finland (Centre of Excellence in Inverse Modelling and Imaging, grant numbers 284715 and 309963)

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We prove a unique continuation property for the fractional Laplacian $ (-\Delta)^s $ when $ s \in (-n/2, \infty)\setminus \mathbb{Z} $ where $ n\geq 1 $. In addition, we study Poincaré-type inequalities for the operator $ (-\Delta)^s $ when $ s\geq 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.

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Mathematics Subject Classification: Primary: 35R30, 46F12; Secondary: 44A12.

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