On quasiconformal maps and semi-linear equations in the plane

• Опубліковано в: : Український математичний вісник
• Дата: 2017
• Автори: Gutlyanskii, V.Y., Nesmelova, O.V., Ryazanov, V.I.
• Формат: Стаття
• Мова: Russian
• Опубліковано: Інститут прикладної математики і механіки НАН України 2017
• Онлайн доступ: https://nasplib.isofts.kiev.ua/handle/123456789/169320
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Цитувати: On quasiconformal maps and semi-linear equations in the plane / V.Y. Gutlyanskii, O.V. Nesmelova, V.I. Ryazanov // Український математичний вісник. — 2017. — Т. 14, № 2. — С. 161-191. — Бібліогр.: 39 назв. — англ.

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.