The inverse Robin boundary value problem in a half-space

Päivärinta, Lassi; Zubeldia, Miren

doi:10.1080/00036811.2014.996556

Cited by 5 publications

(4 citation statements)

References 34 publications

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“…Remark 1.6. In a similar way with results of [17], [18] and subsequent studies of [36], estimates (1.9), (1.11) can be extended to the case when we do not assume that condition (1.5) is fulfiled and consider an appropriate impedance boundary map (Robin-to-Robin map) instead of the Dirichlet-to-Neumann map.…”

Section: )mentioning

confidence: 67%

Effectivized Hölder-logarithmic stability estimates for the Gel’fand inverse problem

Isaev¹,

Novikov²

2014

Inverse Problems

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We give effectivized Hölder-logarithmic energy and regularity dependent stability estimates for the Gel'fand inverse boundary value problem in dimension d = 3. This effectivization includes explicit dependance of the estimates on coefficient norms and related parameters. Our new estimates are given in L 2 and L ∞ norms for the coefficient difference and related stability efficiently increases with increasing energy and/or coefficient difference regularity. Comparisons with preceeding results are given.

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Section: )mentioning

confidence: 67%

Effectivized Hölder-logarithmic stability estimates for the Gel’fand inverse problem

Isaev¹,

Novikov²

2014

Inverse Problems

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“…The classical problem with impedance boundary value condition in the half-space geometry is to understand what kind of surface waves appear on R 3 0 . Related to this, the uniqueness of the solution in many cases requires a special radiation condition [15,26]. In our case it can be shown that the classical Sommerfeld radiation condition (3) ∂ ∂r − ik u(x) = o(|x| −1 ), as |x| → ∞ and uniformly in the sphere x/|x| ∈ S 2 , guarantees the uniqueness for a real-valued and compactly supported λ k .…”

Section: Introductionmentioning

confidence: 72%

Inverse acoustic scattering problem in half-space with anisotropic random impedance

Helin¹,

Lassas²,

Päivärinta³

2017

Journal of Differential Equations

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Abstract. We study an inverse acoustic scattering problem in half-space with a probabilistic impedance boundary value condition. The Robin coefficient (surface impedance) is assumed to be a Gaussian random function with a pseudodifferential operator describing the covariance. We measure the amplitude of the backscattered field averaged over the frequency band and assume that the data is generated by a single realization of λ. Our main result is to show that under certain conditions the principal symbol of the covariance operator of λ is uniquely determined. Most importantly, no approximations are needed and we can solve the full non-linear inverse problem. We concentrate on anisotropic models for the principal symbol, which leads to the analysis of a novel anisotropic spherical Radon transform and its invertibility.

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“…What is more, although there seems to be a burgeoning interest in non-self-adjoint Robin Laplacians in various contexts such as half-spaces [21,22,60] and scattering problems (for example [4,20,48,49] among many others); waveguides (e.g., [14,57,58,59]); thin layers [15,46]; triangles [54,64]; and metric graphs [37], to say nothing of the extensive physics literature on impedance boundary conditions (see for example the references in [21,48,49,58], etc. ), to date many basic spectral properties of this operator on general (bounded) domains seem not yet to have been established.…”

Section: Introductionmentioning

confidence: 99%

On the eigenvalues of the Robin Laplacian with a complex parameter

2022

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We study the spectrum of the Robin Laplacian with a complex Robin parameter α on a bounded Lipschitz domain Ω. We start by establishing a number of properties of the corresponding operator, such as generation properties, analytic dependence of the eigenvalues and eigenspaces on α ∈ C, and basis properties of the eigenfunctions. Our focus, however, is on bounds and asymptotics for the eigenvalues as functions of α: we start by providing estimates on the numerical range of the associated operator, which lead to new eigenvalue bounds even in the case α ∈ R. For the asymptotics of the eigenvalues as α → ∞ in C, in place of the min-max characterisation of the eigenvalues and Dirichlet-Neumann bracketing techniques commonly used in the real case, we exploit the duality between the eigenvalues of the Robin Laplacian and the eigenvalues of the Dirichlet-to-Neumann map. We use this to show that along every analytic curve of eigenvalues, the Robin eigenvalues either diverge absolutely in C or converge to the Dirichlet spectrum, as well as to classify all possible points of accumulation of Robin eigenvalues for large α. We also give a comprehensive treatment of the special cases where Ω is an interval, a hyperrectangle or a ball. This leads to the conjecture that on a general smooth domain in dimension d ≥ 2 all eigenvalues converge to the Dirichlet spectrum if Re α remains bounded from below as α → ∞, while if Re α → −∞, then there is an infinite family of divergent eigenvalue curves, each of which behaves asymptotically like −α 2 .

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The inverse Robin boundary value problem in a half-space

Abstract: Abstract. We study the inverse Robin problem for the Schrödinger equation in a half-space. The potential is assumed to be compactly supported. We first solve the direct problem for dimensions two and three. We then show that the Robin-to-Robin map uniquely determines the potential q.

Cited by 5 publications

References 34 publications

Effectivized Hölder-logarithmic stability estimates for the Gel’fand inverse problem

Effectivized Hölder-logarithmic stability estimates for the Gel’fand inverse problem

Inverse acoustic scattering problem in half-space with anisotropic random impedance

On the eigenvalues of the Robin Laplacian with a complex parameter

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