Geodesic
On the mixed problem for Laplace equation outside cuts, placed along a circumference in a plane
Fundamentalʹnaâ i prikladnaâ matematika, Tome 8 (2002) no. 4, pp. 1129-1158 
A. I. Sgibnev; P. A. Krutitskii. On the mixed problem for Laplace equation outside cuts, placed along a circumference in a plane. Fundamentalʹnaâ i prikladnaâ matematika, Tome 8 (2002) no. 4, pp. 1129-1158. http://geodesic.mathdoc.fr/item/FPM_2002_8_4_a13/
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Voir la notice de l'article provenant de la source Math-Net.Ru

The boundary value problem for harmonic functions outside cuts lying on the arcs of a circumference is considered. The Dirichlet condition is given on one side of each cut and Neumann condition is specified on the other side. The problem is reduced to the Riemann–Hilbert problem for complex analytic function, which is solved in a closed form. An explicit solution of the original problem is obtained.

[1] Krutitskii P. A., “O zadache Rimana–Gilberta i zadache s kosoi proizvodnoi na ploskosti s razrezami vdol okruzhnosti”, Mat. modelirovanie, 2:9 (1990), 114–123 | MR | Zbl

[2] Krutitskii P. A., “O zadache s kosoi proizvodnoi dlya ploskosti s razrezami vdol pryamoi i svyazannykh s nei zadachakh”, Mat. modelirovanie, 2:4 (1990), 143–154 | MR | Zbl

[3] Muskhelishvili N. I., Singulyarnye integralnye uravneniya, Nauka, M., 1968 | MR | Zbl

[4] Krutitskii P. A., “Primenenie potentsialov S. A. Gabova v fizike poluprovodnikov”, ZhVM i MF, 31:1 (1991), 109–121 | MR

[5] Sgibnev A. I., Krutitskii P. A., “Smeshannaya zadacha dlya uravneniya Laplasa vne dugi okruzhnosti”, Vestnik Mosk. un-ta. Ser. 3. Fizika. Astronomiya, 2000, no. 6, 27–30 | MR

[6] Krutitskaya N. Ch., Krutitskii P. A., Malysheva G. Yu., “O dinamike elektronnoi plazmy v zamagnichennoi poluprovodnikovoi plenke s elektrodom, lezhaschim na duge okruzhnosti”, Fundam. i prikl. mat., 2:1 (1996), 147–160 | MR | Zbl

[7] Gosteva A. S., Krutitskaya N. Ch., Krutitskii P. A., “Smeshannaya zadacha v zamagnichennoi poluprovodnikovoi plenke s razrezami vdol pryamoi”, Fundam. i prikl. mat., 6:4 (2000), 1061–1073 | MR | Zbl

[8] Krutitskii P. A., “Yavnoe reshenie zadachi Dirikhle dlya uravneniya sostavnogo tipa v mnogosvyaznoi oblasti”, DAN SSSR, 325:3 (1992), 428–433 | Zbl

[9] Krutitskii P. A., “An initial-boundary value problem for the pseudo-hyperbolic equation of gravity-gyroscopic waves”, Journal of Mathematics of Kyoto University, 37:2 (1997), 343–365 | MR | Zbl

[10] Krutitskii P. A., “An explicit solution of the pseudo-hyperbolic initial problem in a multiply connected region”, Mathematical Methods in the Applied Sciences, 18:11 (1995), 897–925 | DOI | MR | Zbl

[11] Krutitskii P. A., “Stekanie elektricheskogo toka s pryamolineinykh elektrodov v zamagnichennoi poluprovodnikovoi plenke”, ZhVM i MF, 30:11 (1990), 1689–1701 | MR