On the theory of alternating Beltrami equation with many folds

Suppose that, in a simply-connected domain ⊂ C, we are given the differential equation 𝐴(𝑧)𝑓𝑧(𝑧) + 𝐵(𝑧)𝑓𝑧(𝑧) = 0, (𝑧 = 𝑥1 + 𝑖𝑥2 ∈ 𝐷), where 𝐴(𝑧), 𝐵(𝑧) (|𝐴(𝑧)| ̸= |𝐵(𝑧)| a.e. in 𝐷) are finite measurable complexvalued functions. For = 𝜇,𝐵 = −1 the equation is given by the Beltrami equation (see [2, Chapter 2]) 𝑓𝑧(𝑧) = 𝜇(𝑧)𝑓𝑧(𝑧). 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 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 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 𝜇*(𝑧) = −𝐴(𝑧)/𝐵(𝑧) for |𝐴(𝑧)| ≤ |𝐵(𝑧)|, −𝐵(𝑧)/𝐴(𝑧) for |𝐴(𝑧)| > |𝐵(𝑧)|. Below we call this equation associated with (*). Let Γ is finite set of arcs {𝛾} dividing on a subregions {𝐷𝑖}. Let’s designate this partition of area on a subregions 𝑇(Γ). Let’s assume that 𝑇(Γ) supposes a black-white colouring, fix it. We put 𝐸Γ = ⋃︀ 𝛾∈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 ∈ 𝑇(Γ) and each arc ∈ 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 and conformal mapping of the second kind each black region and which is a homeomorphism on each arc ∈ 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 to the equation associated with (*). If for every is holds 𝑓−1 0 ∈ 𝑊1,2 loc (𝑓0(𝐷𝑖)) and 𝑓−1 ∈ 𝑊1,2 loc (𝑓(𝐷𝑖)), where 𝑓−1 is a branch multi-valued function 𝑓−1, then the following hold: 1) 𝑓(𝑓−1 0 (𝑤)) 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 without endpoints is an analytic arcs. In the article the uniqueness theorem for conformal multifolds also is received.