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## Archive of Issues

Russia Moscow
Year
2013
Issue
2
Pages
48-58
 Section Mathematics Title On solutions of third boundary value problem for Laplace equation in a half-infinite cylinder Author(-s) Neklyudov A.V.a Affiliations Bauman Moscow State Technical Universitya 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 $\displaystyle{\int_{\Gamma}}x_1\beta(x)\,dS,$ 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 half-infinite cylinder, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2013, issue 2, pp. 48-58. References Landis E.M., Panasenko G.P. On a variant of theorem of Phragmen–Lindelef type for elliptic equations with coefficients that are periodic in all variables but one, Tr. Semin. Im. I.G. Petrovskogo, 1979, vol. 5, pp. 105–136. Landis E.M., Lakhturov S.S. Behavior at infinity of solutions to elliptic equations that are periodic in all variables but one, Dokl. Akad. Nauk SSSR, 1980, vol. 250, no. 4, pp. 803–806. Oleinik O.A., Iosif’yan G.А. On the behavior at infinity of solutions of second order elliptic equations in domains with noncompact boundary, Mat. Sb., vol. 112, no. 4, pp. 588–610. Oleinik O.A., Yosifian G.А. On the asymptotic behavior at infinity of solutions in linear elasticity, Arch. Ration. Mech. Anal., 1982, vol. 78, no. 1, pp. 29–53. Neklyudov A.V. On the Neumann problem for higher-order divergent elliptic equations in an unbounded domain, close to a cylinder, Tr. Semin. Im. I.G. Petrovskogo, 1991, vol. 16, pp. 192–217. Samaitis K.P. Estimates for solutions of the Neumann and Robin problems for the Laplace equation in a cylinder, Differ. Uravn., 2002, vol. 38, no. 7, pp. 995–996. Samaitis K.P. Some estimates for solutions of the Laplace equation in cylinder-like domains, Differ. Uravn., 2002, vol. 38, no. 8, pp. 1105–1112. Oleinik O.A. Lektsii ob uravneniyakh s chastnymi proizvodnymi (Lectures on partial differential equations), Moscow: Binom, 2005, 260 p. Ladyzhenskaya O.A., Ural’tseva N.N. Lineinye i kvazilineinye uravneniya ellipticheskogo tipa (Linear and quasilinear equations of elliptic type), Moscow: Nauka, 1965, 540 p. Oleinik O.A. On properties of solutions of certain boundary problems for equations of elliptic type, Mat. Sb., 1952, vol. 72, no. 3, pp. 695–702. Full text