The fractional Calderón problem: Low regularity and stability

By Angkana Rüland, Mikko Salo. Nonlinear Analysis, 2017

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Abstract The Calderon problem for the fractional Schrodinger equation was introduced in the work Ghosh et al. (to appear) which gave a global uniqueness result also in the partial data case. This article improves this result in two ways. First, we prove a quantitative uniqueness result showing that this inverse problem enjoys logarithmic stability under suitable a priori bounds. Second, we show that the results are valid for potentials in scale-invariant L p or negative order Sobolev spaces. A key point is a quantitative approximation property for solutions of fractional equations, obtained by combining a careful propagation of smallness analysis for the Caffarelli–Silvestre extension and a duality argument.

Subfields: Computational Theory and Mathematics, Mathematical Physics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/ae51fe41d486baa047575254f683e0f09c4a56e2

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Citations in the corpus

Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems (2020, methodology): “ ball by s-harmonic functions, i.e. functions uwhich satisfy ( ) su= 0 in B 1(0) (see also [27, Remark 7.3]). Theorems 1.5 and 1.6 are proved in section 6. The proofs are almost identical to those in [27, 70] and only slight changes need to be done. We will present the main ideas of the proofs for c”
Getting Acquainted with the Fractional Laplacian (2017, methodology): “See also [55, 79, 80] for different approaches to approximation methods in fractional settings which lead to new proofs, and very refined and quantitative statements.”
Fractional anisotropic Calderón problem on closed Riemannian manifolds (2021, background): “Following this work, inverse problems of recovering lower order terms for fractional elliptic equations have been studied extensively, see for example [15], [14], [35], [34], [1], [4], [5], [6], [7], [8], [9], [30], [31], [33] for some of the important contributions.”
The higher order fractional Calderón problem for linear local operators: Uniqueness (2020, methodology): “1 and section 3) generalizes the study of the uniqueness problem for singular potentials in [36] and the second part (theorem 1.”