The Calderón Problem for a Space-Time Fractional Parabolic Equation

Lai, Ru Yu; Lin, Yi; Rüland, Angkana

doi:10.1137/19m1270288

Cited by 47 publications

(28 citation statements)

References 38 publications

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“…(vi) Is there uniqueness for the conductivity type fractional Calderón problems [10,15] in the higher order cases? (vii) Could recent results on fractional heat equations [50,76] be generalized to the higher order cases? (viii) Does the higher regularity Runge approximation in [11,29] generalize to higher order cases?…”

Section: Proof the Proof Follows Trivially From Properties (1)-(3) Of B Smentioning

confidence: 99%

Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems

Covi¹,

Mönkkönen²,

Railo³

2021

IPI

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We prove a unique continuation property for the fractional Laplacian (−∆) s when s ∈ (−n/2, ∞) \ Z where n ≥ 1. In addition, we study Poincaré-type inequalities for the operator (−∆) s when s ≥ 0. We apply the results to show that one can uniquely recover, up to a gauge, electric and magnetic potentials from the Dirichlet-to-Neumann map associated to the higher order fractional magnetic Schrödinger equation. We also study the higher order fractional Schrödinger equation with singular electric potential. In both cases, we obtain a Runge approximation property for the equation. Furthermore, we prove a uniqueness result for a partial data problem of the d-plane Radon transform in low regularity. Our work extends some recent results in inverse problems for more general operators.

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Section: Proof the Proof Follows Trivially From Properties (1)-(3) Of B Smentioning

confidence: 99%

Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems

Covi¹,

Mönkkönen²,

Railo³

2021

IPI

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“…See [10] for inverse problems for directionally antilocal operators. See [18,21] for inverse problems for linear fractional parabolic operators. See [22] for an inverse problem for a different nonlinear fractional parabolic operator.…”

Section: Introductionmentioning

confidence: 99%

An inverse problem for the fractional porous medium equation

Li¹

2021

Preprint

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“…For non-local operators it is in general rather hard to deduce quantitative stability/ unique continuation bounds of the form (2), as for general nonlocal operators the most commonly used tools in proving quantitative unique continuation -Carleman estimates and frequency function boundsare often not directly available (see however the results in [3,6,10,[14][15][16][17][18][19][20] for qualitative and quantitative bounds for the fractional Laplacian and related operators which can be addressed by adapting local techniques by virtue of the Caffarelli-Silvestre extension). The methods which we present here thus offer alternatives to these arguments for certain classes of nonlocal operators and are based on i. the stability estimates of the (Hausdorff) moment problem, ii.…”

Section: Introductionmentioning

confidence: 99%

On two methods for quantitative unique continuation results for some nonlocal operators

García-Ferrero

Rüland

2020

Communications in Partial Differential Equations

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In this article, we present two mechanisms for deducing logarithmic quantitative unique continuation bounds for certain classes of integral operators. In our first method, expanding the corresponding integral kernels, we exploit the logarithmic stability of the moment problem. In our second method we rely on the presence of branchcut singularities for certain Fourier multipliers. As an application we present quantitative Runge approximation results for the operator L s ðDÞ ¼ P n j¼1 ðÀ@ 2 xj Þ s þ q with s 2 ½ 1 2 , 1Þ and q 2 L 1 acting on functions on R n : ARTICLE HISTORY

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The Calderón Problem for a Space-Time Fractional Parabolic Equation

Cited by 47 publications

References 38 publications

Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems

Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems

An inverse problem for the fractional porous medium equation

On two methods for quantitative unique continuation results for some nonlocal operators

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