Global uniqueness for the fractional semilinear Schrödinger equation

Lai, Ru Yu; Lin, Yi

doi:10.1090/proc/14319

Cited by 45 publications

(36 citation statements)

References 20 publications

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“…in [RS17a], and this stability is optimal [RS18]. Uniqueness for recovering a potential in the anisotropic fractional equation ((−div(A∇u)) s + q)u = 0 was shown in [GLX17], and related inverse problems for the semilinear equation (−∆) s u + q(x, u) = 0 were studied in [LL17]. A reconstruction method for positive potentials based on monotonicity methods was given in [HL17].…”

Section: Introductionmentioning

confidence: 99%

Uniqueness and reconstruction for the fractional Calderón problem with a single measurement

Ghosh

Rüland

Salo

et al. 2020

Journal of Functional Analysis

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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.

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Section: Introductionmentioning

confidence: 99%

Uniqueness and reconstruction for the fractional Calderón problem with a single measurement

Ghosh

Rüland

Salo

et al. 2020

Journal of Functional Analysis

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“…Let us put these results into the context of the literature on the fractional Calderón problem: The problem was first introduced by [GSU16], where the authors treated the case with c ∈ L ∞ (Ω), b = 0 and s ∈ (0, 1), and proved a global uniqueness result for c. For more general nonlocal variable coefficient Schrödinger operators, the fractional Calderón problem was studied in [GLX17]. The techniques based on Runge approximation are strong enough to deal with the case of semilinear equations [LL18] and low regularity, almost critical function spaces for the potential [RS17a]. Even single measurement results are possible [GRSU18] (c.f.…”

Section: Introductionmentioning

confidence: 99%

The Calderón problem for the fractional Schrödinger equation with drift

2020

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We investigate the Calderón problem for the fractional Schrödinger equation with drift, proving that the unknown drift and potential in a bounded domain can be determined simultaneously and uniquely by an infinite number of exterior measurements. In particular, in contrast to its local analogue, this nonlocal problem does not enjoy a gauge invariance. The uniqueness result is complemented by an associated logarithmic stability estimate under suitable apriori assumptions. Also uniqueness under finitely many generic measurements is discussed. Here the genericity is obtained through singularity theory which might also be interesting in the context of hybrid inverse problems. Combined with the results from [GRSU18], this yields a finite measurements constructive reconstruction algorithm for the fractional Calderón problem with drift. The inverse problem is formulated as a partial data type nonlocal problem and it is considered in any dimension n ≥ 1.

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“…Note that both results rely on a very strong unique continuation property, and we will utilize this property from [32] as a key ingredient for our results. Furthermore, for uniqueness results, [30] and [61] solved the Calderón problem for general nonlocal variable elliptic operators and the semilinear case, respectively. In addition, [18] studied the fractional Calderón problem with drift, which shows the global uniqueness result holds for drift and potential simultaneously, which is the first example to demonstrate different results between local and nonlocal inverse problems.…”

Section: Introductionmentioning

confidence: 99%

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

Harrach

Lin²

2020

SIAM J. Math. Anal.

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In this work, we use monotonicity-based methods for the fractional Schrödinger equation with general potentials q ∈ L ∞ (Ω) in a Lipschitz bounded open set Ω ⊂ R n in any dimension n ∈ N. We demonstrate that if-and-only-if monotonicity relations between potentials and the Dirichlet-to-Neumann map hold up to a finite dimensional subspace. Based on these if-and-only-if monotonicity relations, we derive a constructive global uniqueness result for the fractional Calderón problem and its linearized version. We also derive a reconstruction method for unknown obstacles in a given domain that only requires the background solution of the fractional Schrödinger equation, and we prove uniqueness and Lipschitz stability from finitely many measurements for potentials lying in an a-priori known bounded set in a finite dimensional subset of L ∞ (Ω).

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Global uniqueness for the fractional semilinear Schrödinger equation

Cited by 45 publications

References 20 publications

Uniqueness and reconstruction for the fractional Calderón problem with a single measurement

Uniqueness and reconstruction for the fractional Calderón problem with a single measurement

The Calderón problem for the fractional Schrödinger equation with drift

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

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