Optimal stability estimates for a magnetic Schrödinger operator with local data

Potenciano-Machado, Leyter

doi:10.1088/1361-6420/aa7770

Cited by 9 publications

(11 citation statements)

References 28 publications

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“…In dimension two, similar results with full and partial data have been stated in [5,21,22]. We mention also, without being exhaustive, the work of [8,9,15,39,40] dealing with the stability issue associated to this problem and some results inspired by this approach for other PDEs stated in [13,20,29,30,31].…”

Section: Introductionsupporting

confidence: 63%

“…In dimension two, similar results with full and partial data have been stated in [5,22,23]. Moreover, without being exhaustive, we refer to the work of [8,9,15,36,43,44] dealing with the stability issue associated with this problem and some results inspired by this approach for other partial differential equations (PDEs) stated in [13,20,[31][32][33]. Let us remark that all the above-mentioned results have been proved in a bounded domain.…”

Section: Known Resultsmentioning

confidence: 61%

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Recovery of Non-Compactly Supported Coefficients of Elliptic Equations on an Infinite Waveguide

Kian¹

2018

J. Inst. Math. Jussieu

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We consider the unique recovery of a non compactly supported and non periodic perturbation of a Schrödinger operator in an unbounded cylindrical domain, also called waveguide, from boundary measurements. More precisely, we prove recovery of general class of electric potentials from the partial Dirichlet-to-Neumann map, where the Dirichlet data is supported on slightly more than half of the boundary and the Neumann data is taken on the other half of the boundary. We apply this result in different context including recovery of some general class of coefficients from measurements on a bounded subset and recovery of an electric potential, supported on an unbounded cylinder, of a Schrödinger operator in a slab.

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Section: Introductionsupporting

confidence: 63%

Section: Known Resultsmentioning

confidence: 61%

Recovery of Non-Compactly Supported Coefficients of Elliptic Equations on an Infinite Waveguide

Kian¹

2018

J. Inst. Math. Jussieu

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“…This last inverse problem has been studied by [11,13,47,54] and it is strongly connected to the recovery of magnetic Schrödinger operator from boundary measurements which has been intensively studied these last decades. Without being exhaustive, we refer to the work of [9,20,41,55,56,58,60]. In particular, we mention the work of [46] where the recovery of magnetic Schrödinger operators has been addressed for bounded electromagnetic potentials which is the weakest regularity assumption so far for general bounded domains.…”

mentioning

confidence: 99%

Determination of convection terms and quasi-linearities appearing in diffusion equations

Caro,

Kian

2018

Preprint

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We consider the highly nonlinear and ill-posed inverse problem of determining some general expression F (x, t, u, ∇xu) appearing in the diffusion equation ∂tu − ∆xu + F (x, t, u, ∇xu) = 0 on Ω × (0, T ), with T > 0 and Ω a bounded open subset of R n , n 2, from measurements of solutions on the lateral boundary ∂Ω × (0, T ). We consider both linear and nonlinear expression of F (x, t, u, ∇xu). In the linear case, the equation can be seen as a convection-diffusion equation and our inverse problem corresponds to the unique recovery, in some suitable sense, of a time evolving velocity field associated with the moving quantity as well as the density of the medium in some rough setting described by non-smooth coefficients on a Lipschitz domain. In the nonlinear case, we prove the recovery of more general quasi-linear expression appearing in a nonlinear parabolic equation associated with more complex model. Here the goal is to determine the underlying physical low of the system associated with our equation. In this paper, we consider for what seems to be the first time the unique recovery of a general vector valued first order coefficient, depending on both time and space variable. Moreover, we provide results of full recovery of some general class of quasi-linear terms admitting evolution inside the system independently of the solution from measurements at the boundary. These last results improve earlier works of Isakov in terms of generality and precision. In addition, our results give a partial positive answer, in terms of measurements restricted to the lateral boundary, to an open problem posed by Isakov in his classic book (Inverse Problems for Partial Differential Equations) extended to the recovery of quasi-linear terms.

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“…Since then, in [47], the author considered magnetic potentials lying in C 1 , [41] treated the case of magnetic potentials lying in a Dini class and [35] considered this problem with bounded electromagnetic potentials. One of the first results of stability for this problem can be found in [48] and, without being exhaustive, we refer to [4,8,39,40] for some recent improvements of such results and to the works of [1,6,13,36] for the stable recovery of several classes of coefficients appearing in an elliptic equation.…”

Section: Known Resultsmentioning

confidence: 99%

“…More recently, in the specific case of a ball in R 3 , [21] proved the recovery of unbounded magnetic potentials. Concerning results with partial data associated with this last problem, we mention the work of [17,18] and concerning the stability issue, without being exhaustive, we refer to [3,6,7,9,39,40,50]. We mention also the work of [12,22,29] related to problems for hyperbolic and parabolic equations treated with an approach similar to the one considered for elliptic equations.…”

Section: Physical Motivationsmentioning

confidence: 99%

Determination of non-compactly supported electromagnetic potentials in an unbounded closed waveguide

Kian

2019

Rev. Mat. Iberoam.

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We study the inverse problem of determining a magnetic Schrödinger operator in an unbounded closed waveguide from boundary measurements. We consider this problem with a general closed waveguide in the sense that we only require our unbounded domain to be contained into an infinite cylinder. In this context we prove the unique recovery of the magnetic field and the electric potential associated with general bounded and non-compactly supported electromagnetic potentials. By assuming that the electromagnetic potentials are known on the neighborhood of the boundary outside a compact set, we even prove the unique determination of the magnetic field and the electric potential from measurements restricted to a bounded subset of the infinite boundary. Finally, in the case of a waveguide taking the form of an infinite cylindrical domain, we prove the recovery of the magnetic field and the electric potential from partial data corresponding to restriction of Neumann boundary measurements to slightly more than half of the boundary. We establish all these results by mean of a new class of complex geometric optics solutions and of Carleman estimates suitably designed for our problem stated in an unbounded domain and with bounded electromagnetic potentials.

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Optimal stability estimates for a magnetic Schrödinger operator with local data

Cited by 9 publications

References 28 publications

Recovery of Non-Compactly Supported Coefficients of Elliptic Equations on an Infinite Waveguide

Recovery of Non-Compactly Supported Coefficients of Elliptic Equations on an Infinite Waveguide

Determination of convection terms and quasi-linearities appearing in diffusion equations

Determination of non-compactly supported electromagnetic potentials in an unbounded closed waveguide

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