Alexander N. Kondrashov On the uniqueness of solutions of the Beltrami equation with a given real part on a boundary
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https://doi.org/10.15688/mpcm.jvolsu.2023.4.1
Alexander N. Kondrashov
Candidate of Sciences (Physics and Mathematics), Associate Professor, Department of Computer Sciences and Experimental Mathematics,
Volgograd State University
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https://orcid.org/0000-0003-1614-0393
Prosp. Universitetsky, 100, 400062 Volgograd, Russian Federation
Abstract. 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.
Key words: uniqueness theorems, Beltrami equation, complex dilatation, asymptotic behavior, µ-hyperbolic domain, ring-shaped domain.
On the Existence and Uniqueness of a Positive Solution to a Boundary Value Problem for a Nonlinear Ordinary Differential Fourth Order Equation by Kondrashov A.N. is licensed under a Creative Commons Attribution 4.0 International License.
Citation in English: Mathematical Physics and Computer Simulation. Vol. 27 No. 1 2024, pp. 5-16
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