Kondrashov A.N. On the theory of alternating Beltrami equation with many folds

Kondrashov Alеxandеr Nikolaеvich

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


Abstract. Suppose that, in a simply-connected domain D ⊂ C, we are given the differential equation

where A(z), B(z) (|A(z)| ≠ |B(z)| a.e. in D) are finite measurable complexvalued functions. For A = μ, B = −1 the equation is given by the Beltrami equation (see [2, Chapter 2])


It should be noted that (*) was first considered in [16].

    We will call case of the Beltrami equation with |μ(z)| < 1 a.e. in D by classical. The cases |μ(z)| < 1 a.e. in D and |μ(z)| > 1 a.e. in D differ in that, in the first case homeomorphisms do not change sense, and in the second they do. The difference is but formal here. Of interest is the situation when there simultaneously exist subdomains in D in which |μ(z)| < 1 a.e. and subdomains D in which |μ(z)| > 1 a.e. In this case the Beltrami equation is said to be alternating. The problem of the study of alternating Beltrami equations was posed by Volkovyskiˇı [3], and successful progress in this direction was made in [16; 17]. Its solutions are described by mappings with folds, cusps, etc.
    Assign to (*) the classical Beltrami equation with complex dilation


Below we call this equation associated with (*).
    Let Γ is finite set of arcs {γ} dividing D on a subregions {Di}. Let’s designate this partition of area D on a subregions T(Γ). Let’s assume that T(Γ) supposes a black-white colouring, fix it. We put EΓ = Uγ∈Γ[γ] where [γ] is the arc carrier of γ.
    Let f(z) is solution (1) with singularity EΓ in D. Call f(z) which is a homeomorphism on everyone DiT(Γ) and each arc γ ∈ Γ, an (A, B)-multifold.
    Let f(z) is a continuous complex-valued function in D. Call by a conformal multifold the mapping f(z) which is a conformal mapping of the first kind each white region Di and conformal mapping of the second kind each black region Di and which is a homeomorphism on each arc γ ∈ Γ.
    The main result of the article is as follows.

    Theorem. Suppose that there exist an (A, B)-multifold f(z) and a homeomorphic solution f0(z) with singularity EΓ in D to the equation associated with (*). If for every Di is holds  and , where  is a branch multi-valued function f -1, then the following hold:
    1)  is a conformal mapping of the first kind each white region f0(Di) and conformal mapping of the second kind each black region f0(Di);
      2) γ ∈ Γ without endpoints is an analytic arcs.
     In the article the uniqueness theorem for conformal multifolds also is received.

Key words: alternating Beltrami equation, multifolds, conformal multifolds, conformal mapping of the first kind, conformal mapping of the second kind, solution with singularity.

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On the theory of alternating Beltrami equation with many folds 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. №2 (19) 2013 pp. 26-35

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