Geodesic
A generalized Carleman boundary value problem
Sbornik. Mathematics, Tome 11 (1970) no. 1, pp. 25-45 Cet article a éte moissonné depuis la source Math-Net.Ru

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In a finite simply connected domain $D^+$ with a Lyapunov boundary $L$ there is considered the following boundary value problem: to find a function $\Phi^+(z)$ analytic in $D^+$ and $H$-continuous in $D^++L$, according to the boundary condition \begin{equation} \Phi^+[\alpha(t)]=a(t)\Phi^+(t)+b(t)\overline{\Phi^+(t)}+h(t), \end{equation} where $\alpha(t)$ homeomorphically maps $L$ on itself with the preservation $(\alpha=\alpha_+(t))$ or with the change $(\alpha=\alpha_-(t))$ of the direction of the circuit on $L$; $\alpha[\alpha(t)]\equiv t$; $\alpha'(t)\ne0$, $\alpha'(t)\in H(L)$; the functions $a(t),b(t),h(t)\in H(L)$ satisfy the identities \begin{gather*} a(t)a[\alpha(t)]+b(t)\overline{b[\alpha(t)]}=1,\\ a(t)b[\alpha(t)]+\overline{a[\alpha(t)]}b(t)=0,\\ a(t)h[\alpha(t)]+b(t)\overline{h[\alpha(t)]}+h(t)=0. \end{gather*} The Noether theory of problem (1) is constructed, its index is calculated and theorems of its solvability and stability are proved. An investigation of the problem in the case when $\alpha=\alpha_-(t)$ and $|a(t)|>|b(t)|$ is presented. From it there follows when $b(t)\equiv 0$ the known solvability theory of the Carleman problem. Bibliography: 10 titles. 
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     author = {G. S. Litvinchuk and A. P. Nechaev},
     title = {A~generalized {Carleman} boundary value problem},
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     url = {http://geodesic.mathdoc.fr/item/SM_1970_11_1_a1/}
}
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G. S. Litvinchuk; A. P. Nechaev. A generalized Carleman boundary value problem. Sbornik. Mathematics, Tome 11 (1970) no. 1, pp. 25-45. http://geodesic.mathdoc.fr/item/SM_1970_11_1_a1/

[1] F. D. Gakhov, Kraevye zadachi, Fizmatgiz, Moskva, 1963

[2] T. Carleman, “Sur la théorie des equations intégrates et ses applications”, Verhandl. Internat. Math. Kongr. Zürich, 1 (1932), 138–151 | Zbl

[3] D. A. Kveselava, “Nekotorye granichnye zadachi teorii funktsii”, Trudy matem. in-ta AN Gr. SSR, 16 (1948), 39–80 | MR

[4] G. S. Litvinchuk, E. G. Khasabov, “Ob odnom tipe singulyarnykh integralnykh uravnenii”, Sib. matem. zh., 5:3 (1964), 610–627

[5] N. P. Vekua, “Ob odnoi obobschennoi granichnoi zadache Karlemana dlya neskolkikh neizvestnykh funktsii”, Izv. AN SSSR, seriya matem., 20 (1956), 377–384 | MR | Zbl

[6] G. S. Litvinchuk, “Teoriya Nëtera sistemy singulyarnykh integralnykh uravnenii so sdvigom Karlemana i kompleksno sopryazhennymi neizvestnymi”, Izv. AN SSSR, seriya matem., 31 (1967), 563–586 ; 32 (1968), 1414–1417 | Zbl | MR | Zbl

[7] G. S. Litvinchuk, “Ob ustoichivosti odnoi kraevoi zadachi teorii analiticheskikh funktsii”, DAN SSSR, 174:6 (1967), 1268–1270 | Zbl

[8] G. S. Litvinchuk, “Dve teoremy ob ustoichivosti chastnykh indeksov kraevoi zadachi Rimana i ikh prilozhenie”, Izv. VUZov, matematika, 1967, no. 12, 47–57 | Zbl

[9] G. S. Litvinchuk, “Ob indekse i normalnoi razreshimosti odnogo klassa funktsionalnykh uravnenii”, DAN SSSR, 149:5 (1963), 1029–1032 | Zbl

[10] I. B. Simonenko, “Kraevaya zadacha Rimana dlya $n$ par funktsii s izmerimymi koeffitsientami i ee primenenie k issledovaniyu singulyarnykh integralov v prostranstvakh $L_p$ s vesami”, Izv. AN SSSR, seriya matem., 28 (1964), 277–306 | MR