Some harmonic Robin functions in the complex plane

Begehr, Heinrich; Vaitekhovich, Tatyana

doi:10.1515/apam.2010.003

Cited by 21 publications

(5 citation statements)

References 6 publications

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“…The proof of (3.18) for the case of sin α = 0 can be completed proceeding from (3.3), (3.13), (3.20) and estimates of [12] and [3] on G α (x, y, E) for v ≡ 0.…”

Section: )mentioning

confidence: 99%

Stability estimates for recovering the potential by the impedance boundary map

Isaev

Novikov

2013

St. Petersburg Math. J.

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We study the impedance boundary map (or Robin-to-Robin map) for the Schrödinger equation in open bounded domain at fixed energy in multidimensions. We give global stability estimates for determining potential from these boundary data and, as corollary, from the Cauchy data set. Our results include also, in particular, an extension of the Alessandrini identity to the case of the impedance boundary map.

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“…The proof of (3.18) for the case of sin α = 0 can be completed proceeding from (3.3), (3.13), (3.20) and estimates of [12] and [3] on G α (x, y, E) for v ≡ 0.…”

Section: )mentioning

confidence: 99%

Stability estimates for recovering the potential by the impedance boundary map

Isaev

Novikov

2013

St. Petersburg Math. J.

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“…Similar results in the class of inhomogeneous biharmonic and triharmonic functions in the sector were obtained in [9][10][11][12]. Note also that the construction of explicit Green functions of the Robin problem in a circle, when the parameter in the boundary condition is equal to one, is devoted to the work [13,14]. The results of these studies are based on the classical theory of integral representations for analytic, harmonic, and polyharmonic functions on the plane.…”

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confidence: 63%

Green's Functions of Some Boundary Value Problems for Byharmonic Operators and Their Correct Constrictions

Koshanov

Baiarystanov

Daurenkyzy

et al. 2021

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In this paper, a constructive method is given for constructing the Green function of the Dirichlet problem for a biharmonic equation in a multidimensional ball. The need to study boundary value problems for elliptic equations is dictated by numerous practical applications in the theoretical study of the processes of hydrodynamics, electrostatics, mechanics, thermal conductivity, elasticity theory, and quantum physics. The distributions of the potential of the electrostatic field are described using the Poisson equation. When studying the vibrations of thin plates of small deflections, biharmonic equations arise. There are various ways to construct the Green Function of the Dirichlet problem for the Poisson equation. For many types of domains, it is constructed explicitly. And for the Neumann problem in multidimensional domains, the construction of the Green function is an open problem. For the ball, the Green function of the internal and external Neumann problem is constructed explicitly only for the two-dimensional and three-dimensional cases. Finding general correct boundary value problems for differential equations is always an urgent problem. The abstract theory of operator contraction and expansion originates from the work of John von Neumann, in which a method for constructing self-adjoint extensions of a symmetric operator was described and a theory of extension of symmetric operators with finite defect indices was developed in detail. Many problems for partial differential equations lead to operators with infinite defect indices. In the early 80s of the last century, M.O. Otelbaev and his students built an abstract theory that allows us to describe all correct constrictions of a certain maximum operator and separately - all correct extensions of a certain minimum operator, independently of each other, in terms of the inverse operator. In this paper, the correct boundary value problems for the biharmonic operator are described using the Green's function.

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“…Properties (4.4), (4.5) with d ≥ 3, sin α cos α < 0, v ≡ 0 and E = 0 were proven in [29]. For d = 2 see also [5]. In Section 6 we give proofs of (4.4), (4.5) for the case of general α, v and E.…”

Section: Preliminariesmentioning

confidence: 92%

Reconstruction of a Potential From the Impedance Boundary Map

Isaev¹,

Novikov²

2013

EJMCA

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We consider the inverse boundary value problem for the Schrödinger equation at fixed energy with boundary measurements represented as the impedance boundary map (or Robin-to-Robin map). We give formulas and equations for finding (generalized) scattering data for the aforementioned equation from boundary measurements in this impedance representation. Combining these results with results of the inverse scattering theory we obtain efficient methods for reconstructing potential from the impedance boundary map. To our knowledge, results of the present work are new already for the case of Neumannto-Dirichlet map.

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Some harmonic Robin functions in the complex plane

Cited by 21 publications

References 6 publications

Stability estimates for recovering the potential by the impedance boundary map

Stability estimates for recovering the potential by the impedance boundary map

Green's Functions of Some Boundary Value Problems for Byharmonic Operators and Their Correct Constrictions

Reconstruction of a Potential From the Impedance Boundary Map

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