Scattering problems for perturbations of the multidimensional biharmonic operator

Tyni, Teemu; Serov, Valery

doi:10.3934/ipi.2018008

Cited by 34 publications

(57 citation statements)

References 20 publications

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“…In that work the time-dependence in several related scattering coefficients was discussed. This linear operator L 4 has also been studied in higher dimensions [11,13]. In [12] the operator L 4 was generalized by adding a second-order perturbation.…”

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confidence: 99%

Inverse scattering problem for quasi-linear perturbation of the biharmonic operator on the line

Tyni¹,

Serov²

2019

Inverse Problems &Amp; Imaging

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We consider an inverse scattering problem of recovering the unknown coefficients of quasi-linearly perturbed biharmonic operator on the line. These unknown complex-valued coefficients are assumed to satisfy some regularity conditions on their nonlinearity, but they can be discontinuous or singular in their space variable. We prove that the inverse Born approximation can be used to recover some essential information about the unknown coefficients from the knowledge of the reflection coefficient. This information is the jump discontinuities and the local singularities of the coefficients. 2010 Mathematics Subject Classification. Primary: 34L25, 34A55; Secondary: 34L30. ∞ −∞ e − k 2 |y−z| h 1 (z)e i k 2 z dzdy + O(k −5 ) =: J 2 (k) + J 2 (k) + O(k −5 ).

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confidence: 99%

Inverse scattering problem for quasi-linear perturbation of the biharmonic operator on the line

Tyni¹,

Serov²

2019

Inverse Problems &Amp; Imaging

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“…However, the number of contributions concerning time-harmonic problems for infinite Kirchhoff-Love plates at non zero frequencies seems much smaller. From the theoretical point of view, the scattering solutions in the restricted case of purely radial inhomogeneities are analytically computed in [37], while well-posedness in the presence of a potential is rigorously established in [42] for a large enough frequency. From the numerical point of view, some finite element computations with the help of Perfectly Matched Layers can be found in [11].…”

Section: Introductionmentioning

confidence: 99%

On well-posedness of time-harmonic problems in an unbounded strip for a thin plate model

Bourgeois

Chesnel

Fliss

2019

Communications in Mathematical Sciences

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We study the propagation of elastic waves in the time-harmonic regime in a waveguide which is unbounded in one direction and bounded in the two other (transverse) directions. We assume that the waveguide is thin in one of these transverse directions, which leads us to consider a Kirchhoff-Love plate model in a locally perturbed 2D strip. For time harmonic scattering problems in unbounded domains, well-posedness does not hold in a classical setting and it is necessary to prescribe the behaviour of the solution at infinity. This is challenging for the model that we consider and constitutes our main contribution. Two types of boundary conditions are considered: either the strip is simply supported or the strip is clamped. The two boundary conditions are treated with two different methods. For the simply supported problem, the analysis is based on a result of Hilbert basis in the transverse section. For the clamped problem, this property does not hold. Instead we adopt the Kondratiev's approach, based on the use of the Fourier transform in the unbounded direction, together with techniques of weighted Sobolev spaces with detached asymptotics. After introducing radiation conditions, the corresponding scattering problems are shown to be well-posed in the Fredholm sense. We also show that the solutions are the physical (outgoing) solutions in the sense of the limiting absorption principle.

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“…In the case m = 2, we mention that very recently, Tyni and Harju considered inverse backscattering problem for bi‐harmonic operators with lower‐order perturbations and for Schrödinger operators with potentials without compact supports. They are able to recover jumps and singularities of an unknown combination of potentials by using the inverse backscattering Born approximation.…”

Section: Introductionmentioning

confidence: 99%

The inverse backscattering for Schrödinger operators for potentials with noncompact support

Huang

Zheng

et al. 2019

Math Methods in App Sciences

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We consider the inverse backscattering problem for the Schrödinger operator H = −Δ + V on Rn, n ≥ 3, as well as the higher‐order Schrödinger operator ( − Δ)m + V, m = 2,3,…. We show that in some suitable Banach spaces, the map from the potential to the backscattering amplitude is a local diffeomorphism. This kind of problem (for m = 1) was studied by Eskin and Ralston [Comm. Math. Phys., 124(2), 169‐215 (1989)], where they assumed that V∈C0∞. In this paper, we replace the C0∞ assumption on V with certain decay assumption at infinity.

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Scattering problems for perturbations of the multidimensional biharmonic operator

Cited by 34 publications

References 20 publications

Inverse scattering problem for quasi-linear perturbation of the biharmonic operator on the line

Inverse scattering problem for quasi-linear perturbation of the biharmonic operator on the line

On well-posedness of time-harmonic problems in an unbounded strip for a thin plate model

The inverse backscattering for Schrödinger operators for potentials with noncompact support

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