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 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 ∈ R 2 , 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.

On semilinear equations in the complex plane

“…(18), ( ) q f u u = , 0 q > , a particularization of the results in Chapter 1 of [11] shows that a dead core may exist, if and only if 0 1 q < < and λ is large enough. See also the corresponding examples of dead cores in [3]. We have, by Theorem 1, the following: We have also the following consequence of Corollary 1.…”

“…Some definitions and preliminary remarks. Following [3], under a weak solution of Eq. (2), we understand a function…”

“…A fundamental role in the study of the posed problem will play the following factorization theorem (see, e.g., [4], Theorem 1, or [3]…”

“…If, in addition, ϕ is Hölder-continuous, then u is Hölder-continuous in D. Proof. By Theorem 1 in[4] (see, also Theorem 4.1 in[3]), if u is a weak solution of (2), then u U = ω , where ω is a quasiconformal mapping of D onto the unit disk D agreed with A , and U is a weak solution of Eq. (9) with h J = , where J stands for the Jacobian of 1 −…”

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“…In series of our recent papers (see, e.g., [7,8]), we have proposed another application of the theory of quasiconformal mappings to the the study of semilinear partial differential equations of the form…”