On quasiconformal maps and semilinear equations in the plane
- Authors: Gutlyanskiĭ V.1, Nesmelova O.1, Ryazanov V.1
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Affiliations:
- Institute of Applied Mathematics and Mechanics of the NAS of Ukraine
- Issue: Vol 229, No 1 (2018)
- Pages: 7-29
- Section: Article
- URL: https://journals.rcsi.science/1072-3374/article/view/240363
- DOI: https://doi.org/10.1007/s10958-018-3659-6
- ID: 240363
Cite item
Abstract
Assume that Ω is a domain in the complex plane ℂ and A(z) is a symmetric 2×2 matrix function with measurable entries, detA = 1; and such that 1/K|ξ|2 ≤ 〈A(z)ξ, ξ〉 ≤ K|ξ|2, ξ ∈ ℝ2, 1 ≤ K < ∞ . In particular, for semilinear elliptic equations of the form div (A(z)∇u(z)) = f(u(z)) in Ω; we prove a factorization theorem that asserts that every weak solution u to the above equation can be expressed as the composition u = To????; where ???? : Ω → G stands for a K−quasiconformal homeomorphism generated by the matrix function A(z); and T(w) is a weak solution of the semilinear equation ∇T(w) = J(w)f(T(w)) in G: Here, the weight J(w) is the Jacobian of the inverse mapping ????−1: Similar results hold for the corresponding nonlinear parabolic and hyperbolic equations. Some applications of these results to anisotropic media are given.
About the authors
Vladimir Gutlyanskiĭ
Institute of Applied Mathematics and Mechanics of the NAS of Ukraine
Author for correspondence.
Email: vgutlyanskii@gmail.com
Ukraine, Slavyansk
Olga Nesmelova
Institute of Applied Mathematics and Mechanics of the NAS of Ukraine
Email: vgutlyanskii@gmail.com
Ukraine, Slavyansk
Vladimir Ryazanov
Institute of Applied Mathematics and Mechanics of the NAS of Ukraine
Email: vgutlyanskii@gmail.com
Ukraine, Slavyansk