Abstract Green Formulas for Triples of Hilbert Spaces and Sesquilinear Forms

Kopachevsky, Nikolay D.

doi:10.1007/s10958-017-3470-9

Cited by 5 publications

(11 citation statements)

References 7 publications

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“…(1.3) Theorem 1.1 (see [20]). Let a space triple E, F, and G (with scalar products) and a trace operator γ satisfy conditions 1 • -3 • .…”

Section: Abstract Green Formulae For Triples Of Hilbert Spaces and Sesquilinear Formsmentioning

confidence: 99%

“…General approaches applied to investigations of these problems stimulate us to consider them on the basis of the abstract Green formula for triples of Hilbert spaces or sesquilinear forms and on the basis of the corresponding generalized Green formulae for particular spaces arising in problems of mathematical physics (see [14,15,18,20,21]).…”

Section: Introductionmentioning

confidence: 99%

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On Some Problems Generated by a Sesquilinear Form

Kopachevskii

Yakubova

2020

J Math Sci

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Based on the generalized Green formula for a sesquilinear nonsymmetric form for the Laplace operator, we consider spectral nonself-adjoint problems. Several such problems are similar to classical ones; others arise in problems of hydrodynamics and diffraction and in problems with surface dissipation of energy. Properties of solutions of such problems are considered. Also, we study initialboundary value problems generating the considered spectral problems and prove correct solvability theorems for such problems on any interval of time.

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“…(1.3) Theorem 1.1 (see [20]). Let a space triple E, F, and G (with scalar products) and a trace operator γ satisfy conditions 1 • -3 • .…”

Section: Abstract Green Formulae For Triples Of Hilbert Spaces and Sesquilinear Formsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On Some Problems Generated by a Sesquilinear Form

Kopachevskii

Yakubova

2020

J Math Sci

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“…In the present work we employ an abstract approach for studying boundary value problems and transmission problem in Lipschitz domains developed by S.G. Krein and N.D. Kopachevsky. 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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On Oscillations of Two Connected Pendulums Containing Cavities Partially Filled by Incompressible Fluids

Kopachevsky

Voytitsky

Sitshaeva³

2021

J Math Sci

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We consider the linearized problem of small oscillations of two pendulums connected to each other by a spherical hinge. Each pendulum has a cavity partially filled by an incompressible fluid. We study the initial-boundary value problem as well as the corresponding spectral problem on normal motions of the hydromechanical system. We prove theorems on correct solvability of the problem on an arbitrary interval of time both in the case of ideal and viscous fluids in the cavities and we study the corresponding spectral problems as well.

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Abstract Green Formulas for Triples of Hilbert Spaces and Sesquilinear Forms

Cited by 5 publications

References 7 publications

On Some Problems Generated by a Sesquilinear Form

On Some Problems Generated by a Sesquilinear Form

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

On Oscillations of Two Connected Pendulums Containing Cavities Partially Filled by Incompressible Fluids

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