Geodesic
Reflection principle for solutions of the Helmholtz equation in a half-space
Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 12 (2012) no. 1, pp. 102-113 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In this paper we consider the reflection principle for functions satisfying the Helmholtz equation in a half-space. The idea of a proof is to perform continuation of a solution of the Helmholtz equation from an upper half-space into the entire space. It is obtained an analog of Liuvill' theorem. This result states that the function which satisfies the Helmholtz equation (with negative parameter) in the upper half-space, which grows not faster than the power function in the upper half-space and which is equal to zero on a hyperplane is identically equal to zero in the entire space.
Keywords: reflection principle, Helmholtz equation, a half-space. 
@article{VNGU_2012_12_1_a6,
     author = {V. I. Kuzovatov and A. M. Kytmanov},
     title = {Reflection principle for solutions of the {Helmholtz} equation in a~half-space},
     journal = {Sibirskij \v{z}urnal \v{c}istoj i prikladnoj matematiki},
     pages = {102--113},
     year = {2012},
     volume = {12},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VNGU_2012_12_1_a6/}
}
TY  - JOUR
AU  - V. I. Kuzovatov
AU  - A. M. Kytmanov
TI  - Reflection principle for solutions of the Helmholtz equation in a half-space
JO  - Sibirskij žurnal čistoj i prikladnoj matematiki
PY  - 2012
SP  - 102
EP  - 113
VL  - 12
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/VNGU_2012_12_1_a6/
LA  - ru
ID  - VNGU_2012_12_1_a6
ER  - 
%0 Journal Article
%A V. I. Kuzovatov
%A A. M. Kytmanov
%T Reflection principle for solutions of the Helmholtz equation in a half-space
%J Sibirskij žurnal čistoj i prikladnoj matematiki
%D 2012
%P 102-113
%V 12
%N 1
%U http://geodesic.mathdoc.fr/item/VNGU_2012_12_1_a6/
%G ru
%F VNGU_2012_12_1_a6
V. I. Kuzovatov; A. M. Kytmanov. Reflection principle for solutions of the Helmholtz equation in a half-space. Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 12 (2012) no. 1, pp. 102-113. http://geodesic.mathdoc.fr/item/VNGU_2012_12_1_a6/

[1] Stein I., Veis G., Vvedenie v garmonicheskii analiz na evklidovykh prostranstvakh, Per. s angl. V. V. Zharinova, Mir, M., 1974 | Zbl

[2] Colton D. L., Analytic Theory of Partial Differential Equations, Pitman Advanced Publishing Program, 1980 | MR | Zbl

[3] Bitsadze A. V., Uravneniya matematicheskoi fiziki, Nauka, M., 1976 | MR

[4] Smirnov V. I., Kurs vysshei matematiki, v. 4, Nauka, M., 1981

[5] Vladimirov V. S., Uravneniya matematicheskoi fiziki, Nauka, M., 1981 | MR

[6] Kolmogorov A. N., Fomin S. V., Elementy teorii funktsii i funktsionalnogo analiza, Nauka, M., 1981 | MR

[7] Zabreiko P. P., Koshelev A. I., Krasnoselskii M. A. i dr., Integralnye uravneniya, Nauka, M., 1968 | MR

[8] Mikhailov V. P., Differentsialnye uravneniya v chastnykh proizvodnykh, Nauka, M., 1976 | MR

[9] Krylov N. V., Lektsii po ellipticheskim i parabolicheskim uravneniyam v prostranstvakh Geldera, Per. s angl. T. N. Rozhkovskoi, Nauch. kn., Novosibirsk, 1998 | MR | Zbl