# 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:

- 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}

}