6533b85dfe1ef96bd12bddef
RESEARCH PRODUCT
Unique continuation property and Poincar�� inequality for higher order fractional Laplacians with applications in inverse problems
Giovanni CoviJesse RailoKeijo Mönkkönensubject
Pure mathematicsControl and Optimizationfractional Schrödinger equationApproximation propertyPoincaré inequalityRadon transform.01 natural sciencesinversio-ongelmatSchrödinger equationsymbols.namesakefractional Poincaré inequalityOperator (computer programming)Mathematics - Analysis of PDEsFOS: MathematicsDiscrete Mathematics and CombinatoricsUniquenesskvanttimekaniikka0101 mathematicsepäyhtälötMathematicsosittaisdifferentiaaliyhtälötPlane (geometry)inverse problemsComputer Science::Information Retrieval010102 general mathematicsOrder (ring theory)Gauge (firearms)Mathematics::Spectral Theoryunique continuationFunctional Analysis (math.FA)010101 applied mathematicsMathematics - Functional AnalysisModeling and Simulationsymbolsfractional LaplacianAnalysis35R30 46F12 44A12Analysis of PDEs (math.AP)description
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.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2020-01-17 |