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.

This article deals with the weak and strong unique continuation principle for fractional Schrödinger equations with scaling-critical and rough potentials via Carleman estimates. Our methods allow to apply the results to "variable coefficient" versions of fractional Schrödinger equations.

We show global uniqueness in the fractional Calderón problem with a single measurement and with data on arbitrary, possibly disjoint subsets of the exterior. The previous work [GSU16] considered the case of infinitely many measurements. The method is again based on the strong uniqueness properties for the fractional equation, this time combined with a unique continuation principle from sets of measure zero. We also give a constructive procedure for determining an unknown potential from a single exterior measurement, based on constructive versions of the unique continuation result that involve different regularization schemes.This means that zero is not a Dirichlet eigenvalue of (−∆) s + q, and one indeed has a unique solution u ∈ H s (R n ) for any exterior value f . This problem, which was first introduced in [GSU16], should be viewed as a fractional analogue of the classical Calderón problem, which is a well-studied inverse problem for which we refer to the survey article [Uh14] and the references therein. Due to the results of [GSU16], it is known that the Dirichlet-to-Neumann map uniquely determines the potential q, i.e. if q 1 , q 2 ∈ L ∞ (Ω) are such that zero is not a Dirichlet eigenvalue of (−∆) s + q i , i ∈ {1, 2}, thenMoreover, uniqueness holds if the measurements are made on arbitrary, possibly disjoint subsets of the exterior. In [RS17a] this has further been extended to (almost) optimal function spaces, including potentials in L n 2s (Ω). Logarithmic stability for this inverse problem was also proved 2010 Mathematics Subject Classification. 35R11, 35R30, 35J10. Key words and phrases. fractional Calderón problem, single measurement recovery and uniqueness, unique continuation from measurable sets.

Abstract. In this article we present a new strategy of addressing the (variable coefficient) thin obstacle problem. Our approach is based on a (variable coefficient) Carleman estimate. This yields semi-continuity of the vanishing order, lower and uniform upper growth bounds of solutions and sufficient compactness properties in order to carry out a blow-up procedure. Moreover, the Carleman estimate implies the existence of homogeneous blow-up limits along certain sequences and ultimately leads to an almost optimal regularity statement. As it is a very robust tool, it allows us to consider the problem in the setting of Sobolev metrics, i.e. the coefficients are only W 1,p regular for some p > n + 1. These results provide the basis for our further analysis of the free boundary, the optimal (C 1,1/2 -) regularity of solutions and a first order asymptotic expansion of solutions at the regular free boundary which is carried out in a follow-up article, [KRS15], in the framework of W 1,p , p > 2(n + 1), regular coefficients.