В [4] нами был установлен один результат (теорема 3, стр. 108) о допустимой скорости стремления к нулю решений уравнения вида Δu+c(x)u = 0 на концах римановых многообразий с метриками специального вида. Нами установлено, что в двумерном случае этот результат может быть полезен при решении задач несколько иного типа. А именно, нами установлена специальная версия теоремы единственности для уравнения Бельтрами wz = µ(z)wz.

Previously (2019), we established one result on the admissible rate of tending to zero of solutions of an equation of the form Δu + c(x)u = 0 at the ends of Riemannian manifolds with a metric of a special form. In this paper we show that in the two-dimensional case this result can be useful in solving problems of a slightly different type. Namely, for problems in the theory of functions of a complex variable. We have established a special version of the uniqueness theorem for the Beltrami equation ωz = µ(z)wz. Let us present this result. It is known that if ess supD′ |µ(z)| < 1 in each subarea D ′ ⋐ D, then the Beltrami equation has a homeomorphic solution w = f(z) ∈ W 1,2 loc . Moreover, w = f(z) also belongs to the class W 1,2 loc . This solution is unique up to superposition with a conformal mapping. There are two possible cases of f(D) = C or f(D) = G, where G is a simply connected domain whose boundary ∂G has more than two points. In the first case, the domain D is called µ-parabolic, and in the second case it is called µ-hyperbolic. We are only interested in µ-hyperbolic domains. If the domain D is µ-hyperbolic, then it canbe map homeomorphic onto the unit disk B1 = {ζ | ζ ∈ C, |ζ| < 1} using some solution the Beltrami equation. Let us arbitrarily fix ζ = ω(z) = α(z) + iβ(z) such a solution. Let E ⊂ D be a compact subset and F = D ∖E be a ring-shaped domain whose outer part of the boundary coincides with the boundary of ∂D. Let’s introduce the notation Σt = ω−1 (|ω| = t) = {z | z ∈ F, |ω(z)| = 1}, H1(dz) — 1-Hausdorff measure in D. The following theorem is true.