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Title: On quasiconformal maps and semilinear equations in the plane

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|{sup 2} ≤ 〈A(z)ξ, ξ〉 ≤ K|ξ|{sup 2}, ξ ∈ ℝ{sup 2}, 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 = To𝜔; where 𝜔 : 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 𝜔{sup −1}: Similar results hold for the corresponding nonlinear parabolic and hyperbolic equations. Some applications of these results to anisotropic media are given.

Authors:
; ;  [1]
  1. Institute of Applied Mathematics and Mechanics of the NAS of Ukraine (Ukraine)
Publication Date:
OSTI Identifier:
22771561
Resource Type:
Journal Article
Journal Name:
Journal of Mathematical Sciences
Additional Journal Information:
Journal Volume: 229; Journal Issue: 1; Other Information: Copyright (c) 2018 Springer Science+Business Media, LLC, part of Springer Nature; http://www.springer-ny.com; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1072-3374
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICAL METHODS AND COMPUTING; ANISOTROPY; CONFORMAL MAPPING; FACTORIZATION; LAPLACE EQUATION; MATHEMATICAL SOLUTIONS; MATRICES; NONLINEAR PROBLEMS; SYMMETRY

Citation Formats

Gutlyanskiĭ, Vladimir, Nesmelova, Olga, and Ryazanov, Vladimir. On quasiconformal maps and semilinear equations in the plane. United States: N. p., 2018. Web. doi:10.1007/S10958-018-3659-6.
Gutlyanskiĭ, Vladimir, Nesmelova, Olga, & Ryazanov, Vladimir. On quasiconformal maps and semilinear equations in the plane. United States. doi:10.1007/S10958-018-3659-6.
Gutlyanskiĭ, Vladimir, Nesmelova, Olga, and Ryazanov, Vladimir. Thu . "On quasiconformal maps and semilinear equations in the plane". United States. doi:10.1007/S10958-018-3659-6.
@article{osti_22771561,
title = {On quasiconformal maps and semilinear equations in the plane},
author = {Gutlyanskiĭ, Vladimir and Nesmelova, Olga and Ryazanov, Vladimir},
abstractNote = {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|{sup 2} ≤ 〈A(z)ξ, ξ〉 ≤ K|ξ|{sup 2}, ξ ∈ ℝ{sup 2}, 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 = To𝜔; where 𝜔 : 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 𝜔{sup −1}: Similar results hold for the corresponding nonlinear parabolic and hyperbolic equations. Some applications of these results to anisotropic media are given.},
doi = {10.1007/S10958-018-3659-6},
journal = {Journal of Mathematical Sciences},
issn = {1072-3374},
number = 1,
volume = 229,
place = {United States},
year = {2018},
month = {2}
}