On quasiconformal maps and semilinear equations in the plane

Journal Article · Thu Feb 15 04:00:00 EST 2018 · Journal of Mathematical Sciences

DOI:https://doi.org/10.1007/S10958-018-3659-6· OSTI ID:22771561

Gutlyanskiĭ, Vladimir; Nesmelova, Olga; Ryazanov, Vladimir ^[1]

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1. Institute of Applied Mathematics and Mechanics of the NAS of Ukraine (Ukraine)

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.

• OSTI ID: 22771561
• Journal Information: Journal of Mathematical Sciences, Journal Name: Journal of Mathematical Sciences Journal Issue: 1 Vol. 229; ISSN JMTSEW; ISSN 1072-3374
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICS AND COMPUTING
ANISOTROPY
CONFORMAL MAPPING
FACTORIZATION
LAPLACE EQUATION
MATHEMATICAL SOLUTIONS
MATRICES
NONLINEAR PROBLEMS
SYMMETRY