Lipschitz stability for the finite dimensional fractional Calderón problem with finite Cauchy data

Abstract

In this note we discuss the conditional stability issue for the finite dimensional Calderón problem for the fractional Schrödinger equation with a finite number of measurements. More precisely, we assume that the unknown potential q ∈ L ∞ (Ω) in the equation ((−∆) s + q)u = 0 in Ω ⊂ R n satisfies the a priori assumption that it is contained in a finite dimensional subspace of L ∞ (Ω). Under this condition we prove Lipschitz stability estimates for the fractional Calderón problem by means of finitely many Cauchy data depending on q. We allow for the possibility of zero being a Dirichlet eigenvalue of the associated fractional Schrödinger equation. Our result relies on the strong Runge approximation property of the fractional Schrödinger equation.

Monotonicity-Based Inversion of the Fractional Schrödinger Equation II. General Potentials and Stability

“…Note that our Lipschitz stability result in Section 5 complements the result in [71] as we show that any sufficiently high number of measurements (depending only on the a-priori data but not on the unknown potentials) uniquely determines the potential and that Lipschitz stability holds. Moreover, let us stress that the idea of using monotonicity and localized potentials arguments for proving Lipschitz stability (that was already utilized in [21,36,41,72]), differs from traditional approaches that are mostly based on quantitative unique continuation or quantitative Runge approximation, cf., [2,3,4,5,7,9,10,11,12,13,14,19,52,53,56,57,58,65,71,73,78,79]. Our new approach of showing Lipschitz stability seems conceptually simpler as it does not require quantitative analytic estimates.…”

“…Logarithmic stability results for the fractional Schrödinger equation and their optimality were proven by Rüland and Salo in [69,70]. Lipschitz stability for the finite dimensional fractional Calderón problem with a specific set of finitely many measurements (that depend on the unknown potentials) was shown by Rüland and Sincich in [71]. Note that our Lipschitz stability result in Section 5 complements the result in [71] as we show that any sufficiently high number of measurements (depending only on the a-priori data but not on the unknown potentials) uniquely determines the potential and that Lipschitz stability holds.…”

The Calderón problem for the fractional Schrödinger equation with drift

“…Remark 6.6 below on how to avoid this in some cases). However, using arguments as in [RS18b], it is possible to deduce analogous results. Remark 6.6 (Domain monotonicity).…”

“…Stability results for the fractional Calderón problem were first obtained in [RS17a,RS18a], where optimal logarithmic stability estimates had been derived (c.f. also [RS18b] for improvements of this if structural apriori conditions like the finiteness of the underlying function space are satisfied). It is possible to extend the logarithmic estimates to the setting of the fractional Calderón problem with drift.…”

Uniqueness and Lipschitz stability in electrical impedance tomography with finitely many electrodes

“…Together with the recent preprint [41], the present work shows that this idea can also be used to obtain Lipschitz stability estimates, which are usually derived from technically more challenging approaches involving Carleman estimates or quantitative unique continuation, cf. [60,3,56,57,27,6,10,15,65,16,65,64,80,89,90,19,18,74,5,17,14,4,78].…”