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Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2013, Issue 2, Pages 48–58
(Mi vuu376)
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This article is cited in 2 scientific papers (total in 2 papers)
MATHEMATICS
On solutions of third boundary value problem for Laplace equation in a half-infinite cylinder
A. V. Neklyudov Department of Higher Mathematics, Bauman Moscow State Technical University, Moscow, Russia
Abstract:
We study the asymptotic behavior at the infinity of solutions of the Laplace equation in a half-infinite cylinder providing that third boundary value condition is met
$$
\left.{\bigg({{{\partial u
}\over{\partial\nu}}+\beta(x)u}\bigg)}\right|_{\Gamma}=0,
$$
where $\Gamma$ is the lateral surface of the cylinder;
$\beta(x)\geqslant 0$.
We prove that any bounded solution is stabilized to some constant and its
Dirichlet integral is finite. We describe a condition on boundary
coefficient decrease at infinity which provides Dirichlet (dichotomy,
stabilization to zero) or Neumann (trichotomy, stabilization to some
constant which can be nonzero) problem type behavior of solutions. The main
condition on boundary coefficient leading to Dirichlet or Neumann problem
type is established in terms of divergence or convergence correspondingly of
the integral
$\displaystyle{\int_{\Gamma}}x_1\beta(x)\,dS,\quad
$
where the variable $x_1$ corresponds to the direction of an axis of the cylinder.
Keywords:
Laplace equation, third boundary value problem, dichotomy of solutions, trichotomy, stablization.
Received: 11.03.2013
Citation:
A. V. Neklyudov, “On solutions of third boundary value problem for Laplace equation in a half-infinite cylinder”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2013, no. 2, 48–58
Linking options:
https://www.mathnet.ru/eng/vuu376 https://www.mathnet.ru/eng/vuu/y2013/i2/p48
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