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Section  Mathematics 
Title  On solutions of third boundary value problem for Laplace equation in a halfinfinite cylinder 
Author(s)  Neklyudov A.V.^{a} 
Affiliations  Bauman Moscow State Technical University^{a} 
Abstract  We study the asymptotic behavior at the infinity of solutions of the Laplace equation in a halfinfinite 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 $\displaystyle{\int_{\Gamma}}x_1\beta(x)\,dS,$ 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 
UDC  517.956.223 
MSC  35B05, 35J15 
DOI  10.20537/vm130205 
Received  11 March 2013 
Language  Russian 
Citation  Neklyudov A.V. On solutions of third boundary value problem for Laplace equation in a halfinfinite cylinder, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2013, issue 2, pp. 4858. 
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