Geodesic
Alternating Beltrami equation and conformal multifolds
Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 5 (2015), pp. 6-24.

Voir la notice de l'article provenant de la source Math-Net.Ru

The problem of the study of alternating Beltrami equation was posed by L.I. Volkovyskiǐ [5]. In [8] we proved that solutions of the alternating Beltrami equation of a certain structure ($(A,B)$-multifolds) are composition of conformal multifold and suitable homeomorphism. Thus, lines of change of orientation cannot be arbitrary, and only mapped by the specified homeomorphism in analytical arcs. Therefore, understanding of the structure of conformal multifolds is the key to understanding the structure of $(A, B)$-multifolds. The main results of this work. I. The theorem on removability of conformal multifolds cuts. This theorem is about the possibility of extending by continuity from the domain $D_{\Gamma_0} = D\setminus\bigcup_{\gamma\in\Gamma_0}|\gamma|$ to the whole domain $D$. Here $\Gamma_0$ is family of arcs which belong to the set change of type. Theorem 3. Suppose that conditions are hold. (A1) Functions $f_k(z)$ $(k = 1,2)$ are analytical ( antianalytical ) extended from each white ( black ) domain $D_i$ to a domain $\Omega\supset[D]$ and these extensions $f^i_k(z)$ $(i=1,\ldots,N)$, are homeomorphisms of $\Omega$. (A2) $\bigcap_{i=1}^Nf^i_1(\Omega)\supset[f_1(D)]$. Then the conformal multifold $f_2(z)$ in $D_{\Gamma_0}$ is also conformal multifold in $D$. II. Description of a process of constructing conformal multifolds on analytical arcs of change type.
Keywords: alternating Beltrami equation, conformal multifold, black-white cut of domain, multidomain, continuous extending. 
@article{VVGUM_2015_5_a1,
     author = {A. N. Kondrashov},
     title = {Alternating {Beltrami} equation and conformal multifolds},
     journal = {Matemati\v{c}eska\^a fizika i kompʹ\^uternoe modelirovanie},
     pages = {6--24},
     publisher = {mathdoc},
     number = {5},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VVGUM_2015_5_a1/}
}
TY  - JOUR
AU  - A. N. Kondrashov
TI  - Alternating Beltrami equation and conformal multifolds
JO  - Matematičeskaâ fizika i kompʹûternoe modelirovanie
PY  - 2015
SP  - 6
EP  - 24
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/VVGUM_2015_5_a1/
LA  - ru
ID  - VVGUM_2015_5_a1
ER  - 
%0 Journal Article
%A A. N. Kondrashov
%T Alternating Beltrami equation and conformal multifolds
%J Matematičeskaâ fizika i kompʹûternoe modelirovanie
%D 2015
%P 6-24
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/VVGUM_2015_5_a1/
%G ru
%F VVGUM_2015_5_a1
A. N. Kondrashov. Alternating Beltrami equation and conformal multifolds. Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 5 (2015), pp. 6-24. http://geodesic.mathdoc.fr/item/VVGUM_2015_5_a1/

[1] P.\;P. Belinskiy, General Properties of Quasiconformal Mappings, Nauka, Sib. otd-nie Publ., Novosibirsk, 1974, 100 pp. | MR

[2] N. Burbaki, Elements of Mathematics. General Topology. Core Structures, Nauka Publ., M., 1968, 272 pp. | MR

[3] I.\;N. Vekua, Generalized Analytic Functions, Nauka Publ., M., 1988, 512 pp. | MR

[4] M\;A. Krasnoselskiy, A.\;I. Perov, A.\;I. Povolotskiy, P.\;P. Zabreyko, Plane Vector Fields, GIFML Publ., M., 1963, 245 pp. | MR

[5] L.\;I. Volkovyski\v{i}, “Some Problems of the Theory of Quasiconformal Mappings”, Some Problems of Mathematics and Mechanics (to the seventieth birthday of M.\;A. Lavrentyev), Nauka Publ., L., 1970, 128–134 | MR

[6] R. Distel, Graph Theory, Izd-vo IM SO RAN Publ., Novosibirsk, 2002, 241 pp.

[7] A.\;N. Kondrashov, “On the Theory of Degenerate Alternating Beltrami Equations”, Siberian Mathematical Journal, 53:6 (2012), 1321–1337 | MR | Zbl

[8] A.\;N. Kondrashov, “On the Theory of Alternating Beltrami Equation with Many Folds”, Science Journal of Volgograd State University. Mathematics. Physics, 2013, no. 2 (19), 26–35

[9] A.\;N. Kondrashov, “Beltrami Equations with Degenerate on Arcs”, Science Journal of Volgograd State University. Mathematics. Physics, 2014, no. 5 (24), 24–39

[10] M.\;A. Lavrentyev, B.\;V. Shabat, Methods of the Theory of Functions of a Complex Variable, GIFML Publ., M., 1958, 678 pp. | MR

[11] A.\;I. Markushevich, Theory of Analytic Functions, v. 1, Nauka Publ., M., 1967, 488 pp.

[12] U. Srebro, E. Yakubov, “Branched folded maps and alternating Beltrami equations”, Journal d’analyse mathematique, 70 (1996), 65–90 | DOI | MR | Zbl

[13] U. Srebro, E. Yakubov, “Uniformization of maps with folds”, Israel mathematical conference proceedings, 11, 1997, 229–232 | MR | Zbl

[14] U. Srebro, E. Yakubov, “$\mu$-Homeomorphisms”, Contemporary Mathematics AMS, 211, 1997, 473–479 | DOI | MR | Zbl