On quasiconformal maps and semi-linear equations in the plane

Abstract

Assume that Ω is a domain in the complex plane C and A(z) is symmetric 2× 2 matrix function with measurable entries, det A = 1 and such that 1/K|ξ|²≤ 〈A(z)ξ, ξ〉 ≤ K|ξ|², ξ ∊ R², 1 ≤ K < ∞. In particular, for semi-linear elliptic equations of the form div (A(z)∇u(z)) = f(u(z)) in Ω we prove Factorization Theorem that says that every weak solution u to the above equation can be expressed as the composition u = T ◦ ω, 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 semi-linear equation △T(w) = J(w)f(T(w)) in G. Here the weight J(w) is the Jacobian of the inverse mapping ω⁻¹. Similar results hold for the corresponding nonlinear parabolic and hyperbolic equations. Some applications of these results in anisotropic media are given.