A uniqueness result in an inverse hyperbolic problem with analyticity

Anikonov, Yu. E.; Cheng, Jin; Yamamoto, Masahiro

doi:10.1017/s0956792504005649

Cited by 9 publications

(13 citation statements)

References 12 publications

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“…Now we multiply (27) by v(t, x) ∈ H 1 (Q) and integrate over Q. We get, after integrating by parts, taking into account the following: u| Σ = 0, u(T, x) = 0, ∂ ν u| G = 0, u| t=0 = ∂ t u| t=0 = 0 and A (1) is compactly supported in Q:…”

Section: Integral Identitymentioning

confidence: 99%

“…Later it was developed by Belishev, Kurylev, Katchalov, Lassas, Eskin and others; see [17] and references therein. Eskin in [10,12] developed a new approach based on the BC method for determining the time-independent vector and scalar potentials assuming A 0 = 0 in (1). Hyperbolic inverse problems for time-independent coefficients have been extensively studied by Yamamoto and his collaborators as well; see [1,2,8,13,14].…”

Section: Introductionmentioning

confidence: 99%

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An inverse problem for the relativistic Schrödinger equation with partial boundary data

Krishnan

Vashisth

2019

Applicable Analysis

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We study the inverse problem of determining the vector and scalar potentials A(t, x) = (A0, A1, · · · , An) and q(t, x), respectively, in the relativistic Schrödinger equationwhere Ω is a C 2 bounded domain in R n for n ≥ 3 and T > diam(Ω) from partial data on the boundary ∂Q. We prove the unique determination of these potentials modulo a natural gauge invariance for the vector field term.

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Section: Integral Identitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An inverse problem for the relativistic Schrödinger equation with partial boundary data

Krishnan

Vashisth

2019

Applicable Analysis

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“…Hence, by assuming that ∇ • J = 0, we may eliminate a class of non-radiating sources in order to ensure the uniqueness of IP1. Moreover, even though (1.2) can be reduced into a simple vector wave equation under the divergence free condition, we may not use the results in [6] and [23] for the hyperbolic wave equation and Helmholtz equation directly. The reason is that for the Maxwell equation, it is more practical to use the tangential trace of the electric field E(x, t) × ν on the boundary Γ R as a data rather than the total field E(x, t) on the boundary Γ R .…”

Section: Ip1: Determination Of the Spatial Functionmentioning

confidence: 99%

“…In this work, we consider directly how to determine the temporal and spatial source functions for the time-dependent Maxwell equations. We refer to [6,25,34,26,22] for the method of applying Carleman estimates to inverse source problems for hyperbolic systems, and to [32,31] for other formulation of inverse problems which are related to Maxwell's equations. Our approach of converting the time-dependent problem into a multi-frequency problem in the frequency domain seems natural.…”

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confidence: 99%

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Inverse source problems in electrodynamics

Hu¹,

Li²,

Liu³

et al. 2018

Inverse Problems &Amp; Imaging

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Uniqueness to Some Inverse Source Problems for the Wave Equation in Unbounded Domains

Kian

Zhao

2019

Acta Math. Appl. Sin. Engl. Ser.

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This paper is concerned with inverse acoustic source problems in an unbounded domain with dynamical boundary surface data of Dirichlet kind. The measurement data are taken at a surface far away from the source support. We prove uniqueness in recovering source terms of the form f (x)g(t) and f (x 1 , x 2 , t)h(x 3 ), where g(t) and h(x 3 ) are given and x = (x 1 , x 2 , x 3 ) is the spatial variable in three dimensions. Without these a priori information, we prove that the boundary data of a family of solutions can be used to recover general source terms depending on both time and spatial variables. For moving point sources radiating periodic signals, the data recorded at four receivers are prove sufficient to uniquely recover the orbit function. Simultaneous determination of embedded obstacles and source terms was verified in an inhomogeneous background medium using the observation data of infinite time period. Our approach depends heavily on the Laplace transform.problem for doubly hyperbolic equations arising from the nucleation rate reconstruction in the three-dimensional time cone model was analyzed in [32]. A Lipschitz stability result was proved for recovering the spatial component of the source term using interior data and an iterative thresholding algorithm (see also [26] with the final observation data) was tested. However, most of the above mentioned works dealt with recovery of time independent source terms. We refer to [9,36,2,18] where specific time-dependent source terms for hyperbolic equations were considered and to [29] for the recovery of some class of space-time-dependent source terms in the parabolic equation on a wave guide. In the time-harmonic case, inverse source problems with multi-frequency data have been extensively investigated. The increasing stability analysis in recovering spatial-dependent source terms has been carried out from both theoretical and numerical points of view (see e.g., [3,6,4,5,7,11,31,40]).In the time domain, it is very natural to transform the wave scattering problem governed by hyperbolic equations into elliptic inverse problems in the Fourier or Laplace domain with multi-frequency data; see e.g. [24] for determining sound-hard and impedance obstacles in a homogeneous background medium. In [7], the time-domain analysis helps for deriving an increasing stability to time-harmonic inverse source problems via Fourier transform. The same idea was used in [2,20,19] for recovering spatial-dependent sources as well as moving source profiles and orbits in elastodynamics and electromagnetism. The aim of this paper is to analyze the acoustic counterpart with new uniqueness results. Specially, this paper concerns the following four inverse problems with a single boundary surface data:1. Simultaneous determination of sound-soft obstacles and separable source terms in an inhomogeneous medium (Subsection 2.1).2. Simultaneous determination of sound-soft obstacles and general time-dependent source terms from a family of solutions (Subsection 2.2).3. Inverse moving point source pro...

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A uniqueness result in an inverse hyperbolic problem with analyticity

Cited by 9 publications

References 12 publications

An inverse problem for the relativistic Schrödinger equation with partial boundary data

An inverse problem for the relativistic Schrödinger equation with partial boundary data

Inverse source problems in electrodynamics

Uniqueness to Some Inverse Source Problems for the Wave Equation in Unbounded Domains

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