On a model semilinear elliptic equation in the plane
Assume that Ω is a regular domain in the complex plane C and A(z) is symmetric 2 × 2 matrix with measurable entries, det A = 1 and such that 1/K|ξ|² ≤ 〈A(z)ξ, ξ〉 ≤ K|ξ|², ξ ∊ R², 1 ≤ K < ∞. We study the blow-up problem for a model semilinear equation div (A(z)∇u) = e^u in Ω and show that the w...
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| Veröffentlicht in: | Український математичний вісник |
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| Datum: | 2016 |
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| Format: | Artikel |
| Sprache: | English |
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Інститут прикладної математики і механіки НАН України
2016
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/140893 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | On a model semilinear elliptic equation in the plane / V.Y. Gutlyanskii, O.V. Nesmelova, V.I. Ryazanov // Український математичний вісник. — 2016. — Т. 13, № 1. — С. 91-105. — Бібліогр.: 18 назв. — англ. |
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Gutlyanskii, V.Y. Nesmelova, O.V. Ryazanov, V.I. 2018-07-17T17:51:44Z 2018-07-17T17:51:44Z 2016 On a model semilinear elliptic equation in the plane / V.Y. Gutlyanskii, O.V. Nesmelova, V.I. Ryazanov // Український математичний вісник. — 2016. — Т. 13, № 1. — С. 91-105. — Бібліогр.: 18 назв. — англ. 1810-3200 2010 MSC: 30C62, 35J61 https://nasplib.isofts.kiev.ua/handle/123456789/140893 Assume that Ω is a regular domain in the complex plane C and A(z) is symmetric 2 × 2 matrix with measurable entries, det A = 1 and such that 1/K|ξ|² ≤ 〈A(z)ξ, ξ〉 ≤ K|ξ|², ξ ∊ R², 1 ≤ K < ∞. We study the blow-up problem for a model semilinear equation div (A(z)∇u) = e^u in Ω and show that the well-known Liouville–Bieberbach function solves the problem under an appropriate choice of the matrix A(z). The proof is based on the fact that every regular solution u can be expressed as u(z) = T(ω(z)) where ω : Ω → G stands for quasiconformal homeomorphism generated by the matrix A(z) and T is a solution of the semilinear weihted Bieberbach equation ∆T = m(w)e^T in G. Here the weight m(w) is the Jacobian determinant of the inverse mapping ω⁻¹(w). en Інститут прикладної математики і механіки НАН України Український математичний вісник On a model semilinear elliptic equation in the plane Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
On a model semilinear elliptic equation in the plane |
| spellingShingle |
On a model semilinear elliptic equation in the plane Gutlyanskii, V.Y. Nesmelova, O.V. Ryazanov, V.I. |
| title_short |
On a model semilinear elliptic equation in the plane |
| title_full |
On a model semilinear elliptic equation in the plane |
| title_fullStr |
On a model semilinear elliptic equation in the plane |
| title_full_unstemmed |
On a model semilinear elliptic equation in the plane |
| title_sort |
on a model semilinear elliptic equation in the plane |
| author |
Gutlyanskii, V.Y. Nesmelova, O.V. Ryazanov, V.I. |
| author_facet |
Gutlyanskii, V.Y. Nesmelova, O.V. Ryazanov, V.I. |
| publishDate |
2016 |
| language |
English |
| container_title |
Український математичний вісник |
| publisher |
Інститут прикладної математики і механіки НАН України |
| format |
Article |
| description |
Assume that Ω is a regular domain in the complex plane C and A(z) is symmetric 2 × 2 matrix with measurable entries, det A = 1 and such that 1/K|ξ|² ≤ 〈A(z)ξ, ξ〉 ≤ K|ξ|², ξ ∊ R², 1 ≤ K < ∞. We study the blow-up problem for a model semilinear equation div (A(z)∇u) = e^u in Ω and show that the well-known Liouville–Bieberbach function solves the problem under an appropriate choice of the matrix A(z). The proof is based on the fact that every regular solution u can be expressed as u(z) = T(ω(z)) where ω : Ω → G stands for quasiconformal homeomorphism generated by the matrix A(z) and T is a solution of the semilinear weihted Bieberbach equation ∆T = m(w)e^T in G. Here the weight m(w) is the Jacobian determinant of the inverse mapping ω⁻¹(w).
|
| issn |
1810-3200 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/140893 |
| citation_txt |
On a model semilinear elliptic equation in the plane / V.Y. Gutlyanskii, O.V. Nesmelova, V.I. Ryazanov // Український математичний вісник. — 2016. — Т. 13, № 1. — С. 91-105. — Бібліогр.: 18 назв. — англ. |
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AT gutlyanskiivy onamodelsemilinearellipticequationintheplane AT nesmelovaov onamodelsemilinearellipticequationintheplane AT ryazanovvi onamodelsemilinearellipticequationintheplane |
| first_indexed |
2025-12-07T16:10:15Z |
| last_indexed |
2025-12-07T16:10:15Z |
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1850866469739429888 |