On quasiconformal maps and semi-linear equations in the plane
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 Fa...
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| Veröffentlicht in: | Український математичний вісник |
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| Datum: | 2017 |
| Hauptverfasser: | , , |
| Format: | Artikel |
| Sprache: | Russisch |
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Інститут прикладної математики і механіки НАН України
2017
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/169320 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | 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. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862744809456795648 |
|---|---|
| author | Gutlyanskii, V.Y. Nesmelova, O.V. Ryazanov, V.I. |
| author_facet | Gutlyanskii, V.Y. Nesmelova, O.V. Ryazanov, V.I. |
| citation_txt | 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. |
| collection | DSpace DC |
| container_title | Український математичний вісник |
| description | 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.
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| first_indexed | 2025-12-07T20:36:52Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-169320 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1810-3200 |
| language | Russian |
| last_indexed | 2025-12-07T20:36:52Z |
| publishDate | 2017 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Gutlyanskii, V.Y. Nesmelova, O.V. Ryazanov, V.I. 2020-06-10T15:19:24Z 2020-06-10T15:19:24Z 2017 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. 1810-3200 2010 MSC. Primary 30C62, 31A05, 31A20, 31A25, 31B25, 35J61, 35Q15; Secondary 30E25, 31C05, 34M50, 35F45. https://nasplib.isofts.kiev.ua/handle/123456789/169320 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. ru Інститут прикладної математики і механіки НАН України Український математичний вісник On quasiconformal maps and semi-linear equations in the plane Article published earlier |
| spellingShingle | On quasiconformal maps and semi-linear equations in the plane Gutlyanskii, V.Y. Nesmelova, O.V. Ryazanov, V.I. |
| title | On quasiconformal maps and semi-linear equations in the plane |
| title_full | On quasiconformal maps and semi-linear equations in the plane |
| title_fullStr | On quasiconformal maps and semi-linear equations in the plane |
| title_full_unstemmed | On quasiconformal maps and semi-linear equations in the plane |
| title_short | On quasiconformal maps and semi-linear equations in the plane |
| title_sort | on quasiconformal maps and semi-linear equations in the plane |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/169320 |
| work_keys_str_mv | AT gutlyanskiivy onquasiconformalmapsandsemilinearequationsintheplane AT nesmelovaov onquasiconformalmapsandsemilinearequationsintheplane AT ryazanovvi onquasiconformalmapsandsemilinearequationsintheplane |