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

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    * Corresponding author
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.

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


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