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Vestnik Tomskogo Gosudarstvennogo Universiteta. Matematika i Mekhanika, 2019, Number 60, Pages 11–31
DOI: https://doi.org/10.17223/19988621/60/2
(Mi vtgu719)
 

This article is cited in 1 scientific paper (total in 1 paper)

MATHEMATICS

Solution of boundary problems for a two-dimensional elliptic operatordifferential equation in an abstract Hilbert space using the method of boundary integral equations

Ivanov D.Yu.

Moscow State University of Railway Engeneering (MIIT), Moscow, Russian Federation
Full-text PDF (622 kB) Citations (1)
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Abstract: In this paper, we study boundary-value problems of the first, second, and third kinds for the differential-operator equation $\Delta_2\mathbf{u} = \mathbf{Bu}$ ($\Delta_2\equiv\partial_{x_1x_1}^2+\partial_{x_2x_2}^2$) in an open two-dimensional bounded simply connected domain $\Omega^+$ or its open exterior $\Omega^-$. Here, $\mathbf{u}(x_1,x_2)$ is a vector function with values in an abstract Hilbert space $H$; $\mathbf{B}$ is a linear closed densely operator defined in the space $H$ and generating an exponentially decreasing $C_0$-semigroup of contractions $\mathbf{T}(\tau)$: $||\mathbf{T}(\tau)||\leqslant \exp(-p\tau)$ ($p > 0$). Solutions of the boundary-value problems are obtained in the form of vector potentials with unknown vector functions similar to density functions, which are found from Fredholm boundary integral equations of the second kind, wherein kernels of integral operators are expressed through the $C_0$-semigroup $\mathbf{T}(\tau)$. Let $\partial\Omega$ be the boundary of the domain $\Omega^{\pm}$. Under the condition $\partial\Omega\in C^2$, the stable solvability of the boundary-value problems in the space $C(\overline{\Omega^{\pm}};H)$ is proved. Here, $C(\overline{\Omega^{\pm}};H)$ is the Banach space of vector functions, continuous on the closed set $\overline{\Omega^{\pm}}$ with values in the space $H$. The stable solvability of the boundary integral equations in the spaces $L_2(\partial\Omega; H)$ and $C^k(\partial\Omega; H^n_{\mathbf{B}})$ ($k, n\geqslant0$) is also proved under the conditions $\partial\Omega\in C^2$ and $\partial\Omega\in C^{k+2}$, respectively. Here, $L_2(\partial\Omega; H)$ is the Hilbert space of vector functions, square-summable on the set $\partial\Omega$ with values in the space $H$; $C^k(\partial\Omega; H^n_{\mathbf{B}})$ is the Banach space of vector functions, $k$ times continuously differentiable on the set $\partial\Omega$ with values in the Sobolev type space $H^n_{\mathbf{B}}$ defined by powers $n+1$ of the operator $\mathbf{B}$.
Keywords: Boundary-value problem, differential-operator equation, boundary integral equation, semigroup of operators, generator, vector-valued function, operator-valued function, unitary dilation.
Received: 15.02.2018
Bibliographic databases:
Document Type: Article
UDC: 517.956.2, 517.968.25
MSC: 31A25, 34G10, 35J25
Language: Russian
Citation: Ivanov D.Yu., “Solution of boundary problems for a two-dimensional elliptic operatordifferential equation in an abstract Hilbert space using the method of boundary integral equations”, Vestn. Tomsk. Gos. Univ. Mat. Mekh., 2019, no. 60, 11–31
Citation in format AMSBIB
\Bibitem{Iva19}
\by Ivanov~D.Yu.
\paper Solution of boundary problems for a two-dimensional elliptic operatordifferential equation in an abstract Hilbert space using the method of boundary integral equations
\jour Vestn. Tomsk. Gos. Univ. Mat. Mekh.
\yr 2019
\issue 60
\pages 11--31
\mathnet{http://mi.mathnet.ru/vtgu719}
\crossref{https://doi.org/10.17223/19988621/60/2}
\elib{https://elibrary.ru/item.asp?id=39386752}
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  • This publication is cited in the following 1 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Вестник Томского государственного университета. Математика и механика
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