Geodesic
The Poisson integral and Green's function for one strongly elliptic system of equations in a circle and an ellipse
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 56 (2016) no. 12, pp. 2065-2072 
A. O. Bagapsh. The Poisson integral and Green's function for one strongly elliptic system of equations in a circle and an ellipse. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 56 (2016) no. 12, pp. 2065-2072. http://geodesic.mathdoc.fr/item/ZVMMF_2016_56_12_a6/
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Voir la notice de l'article provenant de la source Math-Net.Ru

For a strongly elliptic system of second-order equations of a special form, formulas for the Poisson integral and Green's function in a circle and an ellipse are obtained. The operator under consideration is represented by the sum of the Laplacian and a residual part with a small parameter, and the solution to the Dirichlet problem is found in the form of a series in powers of this parameter. The Poisson formula is obtained by the summation of this series.

[1] Bitsadze A. V., “O edinstvennosti resheniya zadachi Dirikhle dlya ellipticheskikh uravnenii s chastnymi proizvodnymi”, Uspekhi matem. nauk, 3:6 (1948), 212–212

[2] Bitsadze A. V., Uravneniya smeshannogo tipa, Izd-vo AN SSSR, M., 1959

[3] Schechter M., “Various types of boundary conditions of elliptic equations”, Commun. Pure Appl. Math., 13 (1960), 407–425 | DOI | MR | Zbl

[4] Chineese Math., 1 (1960), 288–299 | Zbl

[5] Agmon S., Douglis A., Nirenberg L., “Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions”, Comm. Pure Appl. Math., 17 (1964), 35–92 | DOI | MR | Zbl

[6] Hua Loo-Keng, Lin Wei, Wu Ci-Quian, Second-order systems of partial differential equations in the plane, Pitman Advanced Publishing Program, Boston–London–Melbourne, 1985 | MR | Zbl

[7] Verchota G. C., Vogel A. L., “Nonsymmetric systems on nonsmooth planar domains”, Transactions of The American Mathematical Society, 349:11 (1997), 4501–4535 | DOI | MR | Zbl

[8] Petrovskii I. G., Lektsii ob uravneniyakh s chastnymi proizvodnymi, Fizmatgiz, M., 1961

[9] Somigliana C., “Sui sisteme simmetrici di equazioni a derivate parziali”, Ann. Mat. Pure ed Appl., II, 22 (1894), 143–156 | DOI

[10] Vishik M. I., “O silno ellipticheskikh sistemakh differentsialnykh uravnenii”, Matem. sbornik, 29 (1951), 615–676 | Zbl

[11] Markushevich A. I., Teoriya analiticheskikh funktsii, v. 1, Nauka, M., 1967

[12] Lavrentev M. A., Shabat B. V., Metody teorii funktsii kompleksnogo peremennogo, Nauka, M., 1965

[13] Shabat B. V., Vvedenie v kompleksnyi analiz, v. 1, Funktsii odnogo peremennogo, Nauka, M., 1985