Multicomponent conjugation problems and auxiliary abstract boundary-value problems

Voititsky, V. I.; Kopachevsky, Nikolay D.; Starkov, P. A.

doi:10.1007/s10958-010-0078-8

Cited by 9 publications

(4 citation statements)

References 15 publications

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“…Orazov in [29] established a relation between the Jordan chains of the pencil and the operator root in the case of real eigenvalues. It was proved there that in the general case the entire discrete spectrum of the pencil ( ) splits into the four parts (21) where the sets ⊂ R, = 1, 2, can have non-empty intersection and…”

Section: Theorems On Completeness Of Root Functionsmentioning

confidence: 99%

“…It is based on using abstract and generalized Green's formula adapted for a particular problem (see works [18]- [23]). In particular, such approach was used in works by the second co-author of the paper (see [21], [24], [25]) in studying some problems with the parameter linearly involved in an equation and boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

“…In order to formulate problem (1)-(3) in operator form, we employ a generalized Green's formula for the Laplace operator and mixed boundary conditions, which arises as a particular case of the abstract Green's formula proved in works by N.D. Kopachevsky with co-authors [21]- [23]. In the last work the following result was provided.…”

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confidence: 99%

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On spectral properties of one boundary value problem with a surface energy dissipation

Андронова¹,

Войтицкий²

2017

Ufimskii Matematicheskii Zhurnal

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Abstract. We study a spectral problem in a bounded domain Ω ⊂ R depending on a bounded operator coefficient > 0 and a dissipation parameter > 0. In the general case we establish sufficient conditions ensuring that the problem has a discrete spectrum consisting of countably many isolated eigenvalues of finite multiplicity accumulating at infinity. We also establish the conditions, under which the system of root elements contains an Abel-Lidskii basis in the space 2 (Ω). In model one-and two-dimensional problems we establish the localization of the eigenvalues and find critical values of .Keywords: spectral parameter, quadratic operator pencil, localization of eigenvalues, compact operator, Schatten-von-Neumann classes , Abel-Lidskii basis property. Mathematics Subject Classification: 35P05, 35P10To the memory of Tomas Yakovlevich Azizov, whose results and lectures helped the authors in writing this work. IntroductionIn this work we study the spectral properties of one linear problem in mathematical physics depending on the dimension of a domain Ω ⊂ R (with a piece-wise smooth boundary Ω), a bounded in 2 (Ω) operator coefficient > 0 and a parameter > 0 modeling the intensity of the energy dissipation on a part of the boundary Γ. Namely, we study the problem= 0 (on := Ω ∖ Γ),for an unknown field = ( ) ( ∈ Ω) and a spectral parameter ∈ C.This spectral problem is generated by the initial boundary value problem arising as a lin-

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Section: Theorems On Completeness Of Root Functionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On spectral properties of one boundary value problem with a surface energy dissipation

Андронова¹,

Войтицкий²

2017

Ufimskii Matematicheskii Zhurnal

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“…The results clearly can be generalized to the case in which the surface is divided into finitely many domains by a finite set of Lipschitz (n − 2)-dimensional closed surfaces without self-intersections and without common points (we prefer to make this assumption to be careful) and either the Dirichlet condition, or the Neumann condition, or the spectral condition is posed in each of these domains (cf. [34], [23], and [50]). …”

Section: (72)mentioning

confidence: 99%

Mixed problems in a Lipschitz domain for strongly elliptic second-order systems

Agranovich

2011

Funct Anal Its Appl

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We consider mixed problems for strongly elliptic second-order systems in a bounded domain with Lipschitz boundary in the space R n . For such problems, equivalent equations on the boundary in the simplest L 2 -spaces H s of Sobolev type are derived, which permits one to represent the solutions via surface potentials. We prove a result on the regularity of solutions in the slightly more general spaces H s p of Bessel potentials and Besov spaces B s p . Problems with spectral parameter in the system or in the condition on a part of the boundary are considered, and the spectral properties of the corresponding operators, including the eigenvalue asymptotics, are discussed.

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On Some Class of Self-adjoint Boundary Value Problems with the Spectral Parameter in the Equations and the Boundary Conditions

Voytitsky

2012

Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations

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Multicomponent conjugation problems and auxiliary abstract boundary-value problems

Cited by 9 publications

References 15 publications

On spectral properties of one boundary value problem with a surface energy dissipation

On spectral properties of one boundary value problem with a surface energy dissipation

Mixed problems in a Lipschitz domain for strongly elliptic second-order systems

On Some Class of Self-adjoint Boundary Value Problems with the Spectral Parameter in the Equations and the Boundary Conditions

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