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arXiv · 2001.06210

Unique continuation property and Poincar\'e inequality for higher order fractional Laplacians with applications in inverse problems

Giovanni Covi·Keijo Mönkkönen·Jesse Railo
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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 Poincar\'e-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 Schr\"odinger equation. We also study the higher order fractional Schr\"odinger 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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Giovanni Covi, Keijo Mönkkönen, Jesse Railo. 2020-01-17. Unique continuation property and Poincar\'e inequality for higher order fractional Laplacians with applications in inverse problems. https://doi.org/10.3934/ipi.2021009

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