Kondrashov A.N. Alternating Beltrami Equation and Conformal Multifolds

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http://dx.doi.org/10.15688/jvolsu1.2015.5.1

Kondrashov Alexander Nikolaеvich 

Candidate of Physical and Mathematical Sciences, Associate Professor,
Department of Computer Sciences and Experimental Mathematics,
Volgograd State University
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Prosp. Universitetsky, 100, 400062 Volgograd, Russian Federation

 
Abstract. The problem of the study of alternating Beltrami equation was posed by L.I. Volkovyskii [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  DΓ0    = D \ ∪γ∈ Γ0 |γ| to the whole domain D. Here Γ0 is family of arcs which belong to the set change of type.
Theorem 3. Suppose that conditions are hold.
(A1) Functions fk(z) (k = 1, 2) are analytical ( antianalytical ) extended from each white ( black ) domain Di to a domain Ω ⊃ [D] and these extensions fki (z) (i = 1, . . . ,N), are homeomorphisms of Ω.
(A2) ∩i=1N f1i(Ω) ⊃ [f1(D)].
Then the conformal multifold f2(z) in DΓ0 is also conformal multifold in D.
II. Description of a process of constructing conformal multifolds on analytical arcs of change type.
 
 
Key words: alternating Beltrami equation, conformal multifold, black-white cut of domain, multidomain, continuous extending.

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Alternating Beltrami Equation and Conformal Multifolds by Kondrashov A.N. is licensed under a Creative Commons Attribution 4.0 International License.

Citation in English: Science Journal of Volgograd State University. Mathematics. Physics. №5 (30) 2015 pp. 6-24

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