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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
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
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
Linking options:
https://www.mathnet.ru/eng/vtgu719 https://www.mathnet.ru/eng/vtgu/y2019/i60/p11
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Abstract page: | 197 | Full-text PDF : | 51 | References: | 28 |
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