The fractional Calderón problem: Low regularity and stability

Abstract

The Calderón problem for the fractional Schrödinger equation was introduced in the work [GSU16], 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.

An inverse problem for the fractional Schrödinger equation in a magnetic field

“…If this implies that u = 0 in Ω as well, we say that FMSE has got the WUCP. It is known that WUCP holds if both A and q vanish, that is, in the case of the fractional Laplace equation (see [37]).…”

“…The well-posedness of the direct problem is granted by the assumption that 0 is not an eigenvalue for the left hand side of FMSE (see e.g. [37]). We can therefore define the DN map Λ s A,q : H s (Ω e ) → (H s (Ω e )) * from the bilinear form associated to the equation.…”

Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators

“…Recently, Rüland and Salo [30] studied the fractional Calderón problem under lower regularity conditions and established the stability results for the determination of unknown potentials. They [29] proved the optimal logarithmic stability for the corresponding inverse problem associated with the fractional Schrödinger equation.…”

The Calderón problem for the fractional magnetic operator

“…This result was later generalized in [6]. See [1,2,7,16] for more results related with the Calderón Problem for fractional operators.…”