Geodesic
Asymptotics of the solution of a boundary value problem for Poisson's equation in the exterior of a cylinder and a half-cylinder
Sbornik. Mathematics, Tome 46 (1983) no. 2, pp. 183-202 
A. M. Il'in. Asymptotics of the solution of a boundary value problem for Poisson's equation in the exterior of a cylinder and a half-cylinder. Sbornik. Mathematics, Tome 46 (1983) no. 2, pp. 183-202. http://geodesic.mathdoc.fr/item/SM_1983_46_2_a3/
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     author = {A. M. Il'in},
     title = {Asymptotics of the solution of a boundary value problem for {Poisson's} equation in the exterior of a cylinder and a half-cylinder},
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     url = {http://geodesic.mathdoc.fr/item/SM_1983_46_2_a3/}
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An asymptotic expansion of the solution of the equation $\Delta u=f(x)$, where the function $f(x)$ tends to zero as $|x|\to\infty$ faster than any power of $|x|^{-1}$, is constructed for the second boundary value problem in the exterior of a cylinder and a half-cylinder in three-dimensional space. The asymptotic expression as $|x|\to\infty$ is constructed with accuracy to any power of $|x|^{-1}$ uniformly with respect to all directions. The Fourier transform in the coordinate along the axis of the cylinder is used for the cylinder, and the asymptotic behavior of the Green function is determined along the way. The results obtained for the cylinder with subsequent application of the method of asymptotic successive approximations are used in the case of the boundary value problem for the half-cylinder. Bibliography: 7 titles.

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[5] Muravei L. A., “Asimptoticheskoe povedenie pri bolshikh znacheniyakh vremeni reshenii vtoroi i tretei vneshnikh kraevykh zadach dlya volnovogo uravneniya s dvumya prostranstvennymi peremennymi”, Tr. Matem. in-ta im. V. A. Steklova, 126 (1973), 73–144 | MR | Zbl

[6] Brychkov Yu. A., Prudnikov A. P., Integralnye preobrazovaniya obobschennykh funktsii, Nauka, M., 1977 | MR | Zbl

[7] Ilin A. M., “Kraevaya zadacha dlya ellipticheskogo uravneniya vtorogo poryadka v oblasti s uzkoi schelyu. 2. Oblast s malym otverstiem”, Matem. sb., 103 (145) (1977), 265–284 | MR