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Continuation

Inverse Problems and Imaging

<p style='text-indent:20px;'>We prove a unique continuation property for the fractional Laplacian <inline-formula><tex-math id="M1111">\begin{document}$ (-\Delta)^s $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M2">\begin{document}$ s \in (-n/2, \infty)\setminus \mathbb{Z} $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M3">\begin{document}$ n\geq 1 $\end{document}</tex-math></inline-formula>. In addition, we study Poincaré-type inequalities for the operator <inline-formula><tex-math id="M4">\begin{document}$ (-\Delta)^s $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M5">\begin{document}$ s\geq 0 $\end{document}</tex-math></inline-formula>. We apply the results to show that one can uniquely recover, up to a gauge, electric and magnetic potentials from the Dirichlet-to-Neumann map associated to the higher order fractional magnetic Schrödinger equation. We also study the higher order fractional Schrödinger equation with singular electric potential. In both cases, we obtain a Runge approximation property for the equation. Furthermore, we prove a uniqueness result for a partial data problem of the <inline-formula><tex-math id="M6">\begin{document}$ d $\end{document}</tex-math></inline-formula>-plane Radon transform in low regularity. Our work extends some recent results in inverse problems for more general operators.

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