A1 Journal article (refereed)
Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems (2021)

Covi, G., Mönkkönen, K., & Railo, J. (2021). Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems. Inverse Problems and Imaging, 15(4), 641-681. https://doi.org/10.3934/ipi.2021009

JYU authors or editors

Publication details

All authors or editors: Covi, Giovanni; Mönkkönen, Keijo; Railo, Jesse

Journal or series: Inverse Problems and Imaging

ISSN: 1930-8337

eISSN: 1930-8345

Publication year: 2021

Volume: 15

Issue number: 4

Pages range: 641-681

Publisher: American Institute of Mathematical Sciences (AIMS)

Publication country: United States

Publication language: English

DOI: https://doi.org/10.3934/ipi.2021009

Publication open access: Other way freely accessible online

Publication channel open access: 

Publication is parallel published (JYX): https://jyx.jyu.fi/handle/123456789/78405

Web address of parallel published publication (pre-print): https://arxiv.org/abs/2001.06210

Abstract

We prove a unique continuation property for the fractional Laplacian (−Δ)s when s∈(−n/2,∞)∖Z where n≥1. In addition, we study Poincaré-type inequalities for the operator (−Δ)s when s≥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.

Keywords: inverse problems; partial differential equations; inequalities (mathematics); quantum mechanics

Free keywords: inverse problems; unique continuation; fractional Laplacian; fractional Schrödinger equation; fractional Poincaré inequality; Radon transform.

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Ministry reporting: Yes

Reporting Year: 2021

JUFO rating: 2