On corner scattering for operators of divergence form and applications to inverse scattering

Cakoni, Fioralba; Xiao, Jingni

doi:10.1080/03605302.2020.1843489

Cited by 32 publications

(32 citation statements)

References 25 publications

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“…The relevant properties have also been extensively and intensively studied for acoustic and elastic scattering problems. We refer to [9][10][11][12][13][14][15][16][17][18] for related studies which are concerned more on the non-scattering aspect, [2][3][4]12,[19][20][21][22][23][24] for the related studies which are concerned more with the local and global geometric properties of the relevant transmission eigenfunctions, and [4,12,13,16,20,25,26] for related applications to inverse problems, as well as a recent survey paper [27]. Finally, we would like to mention in passing the related studies on the transmission eigenvalues; see [28,29] and the references cited therein.…”

Section: Physical Background and Discussionmentioning

confidence: 99%

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On vanishing and localizing around corners of electromagnetic transmission resonances

Diao

Liu

Wang

et al. 2021

Partial Differ. Equ. Appl.

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Section: Physical Background and Discussionmentioning

confidence: 99%

“…It can be directly verified that the electric Herglotz wave function E g given by (2. 16) is an entire solution to…”

Section: 4)mentioning

confidence: 99%

On vanishing and localizing around corners of electromagnetic transmission resonances

Diao

Liu

Wang

et al. 2021

Partial Differ. Equ. Appl.

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“…The local geometric property was first discovered and investigated in [5,6], and was further studied in [2][3][4]9,19,[22][23][24] for different geometric and physical setups. We also refer to [2,4,[7][8][9][10][12][13][14]22,23,30,33,34] for related studies in characterizing non-scattering waves (locally) around corner/singular points. The global geometric property was first discovered and investigated in [15], and was further studied in [16,20,21] for different geometric and physical setups.…”

Section: Introductionmentioning

confidence: 99%

Boundary localization of transmission eigenfunctions in spherically stratified media

Jiang¹,

Liu²,

Zhang³

et al. 2022

Preprint

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Consider the transmission eigenvalue problem for u ∈ H 1 (Ω) and v ∈ H 1 (Ω):where Ω is a ball in R N , N = 2, 3. If σ and n are both radially symmetric, namely they are functions of the radial parameter r only, we show that there exists a sequence of transmission eigenfunctions {um, vm} m∈N associated with km → +∞ as m → +∞ such that the L 2 -energies of vm's are concentrated around ∂Ω. If σ and n are both constant, we show the existence of transmission eigenfunctions {uj, vj} j∈N such that both uj and vj are localized around ∂Ω. Our results extend the recent studies in [15,16]. Through numerics, we also discuss the effects of the medium parameters, namely σ and n, on the geometric patterns of the transmission eigenfunctions.

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“…Related results concerning different models including the same operator but in the space H n , the same operator with conductive boundary conditions, similar problems for Maxwell equations and for the Lamé system, can be found in [3,5,8,14,17], etc. In [11], for the first time corner scattering for the operator ∇ • γ∇ + k 2 ρ is considered, and corner scattering results are shown for convex corners in R 2 .…”

Section: Introductionmentioning

confidence: 99%

A New Type of CGO Solutions and its Applications in Corner Scattering

Xiao

2021

Preprint

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We consider corner scattering for the operator ∇ • γ(x)∇ + k 2 ρ(x) in R 2 , with γ a positive definite symmetric matrix and ρ a positive scalar function. A corner is referred to one that is on the boundary of the (compact) support of γ(x) − I or ρ(x) − 1, where I stands for the identity matrix. We show that any admissible incident field will be scattered by such corners, which are allowed to be concave. Moreover, we provide a brief discussion on the existence of non-scattering waves when γ − I has a jump across the corner. In order to prove the results, we construct a new type of complex geometric optics (CGO) solutions.

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On corner scattering for operators of divergence form and applications to inverse scattering

Cited by 32 publications

References 25 publications

On vanishing and localizing around corners of electromagnetic transmission resonances

On vanishing and localizing around corners of electromagnetic transmission resonances

Boundary localization of transmission eigenfunctions in spherically stratified media

A New Type of CGO Solutions and its Applications in Corner Scattering

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