August  2021, 15(4): 641-681. doi: 10.3934/ipi.2021009

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

Received  March 2020 Revised  September 2020 Published  January 2021

Fund Project: 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)

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.

Citation: Giovanni Covi, Keijo Mönkkönen, Jesse Railo. Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems. Inverse Problems & Imaging, 2021, 15 (4) : 641-681. doi: 10.3934/ipi.2021009
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show all references

References:
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[13]

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A. D'Agnolo and M. Eastwood, Radon and Fourier transforms for $\mathcal{D}$-modules, Adv. Math., 180 (2003), 452-485.  doi: 10.1016/S0001-8708(03)00011-2.  Google Scholar

[18]

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Q. DuM. GunzburgerR. B. Lehoucq and K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Rev., 54 (2012), 667-696.  doi: 10.1137/110833294.  Google Scholar

[20]

Q. DuM. GunzburgerR. B. Lehoucq and K. Zhou, A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws, Math. Models Methods Appl. Sci., 23 (2013), 493-540.  doi: 10.1142/S0218202512500546.  Google Scholar

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[23]

M. M. Fall and V. Felli, Unique continuation property and local asymptotics of solutions to fractional elliptic equations, Comm. Partial Differential Equations, 39 (2014), 354-397.  doi: 10.1080/03605302.2013.825918.  Google Scholar

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