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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| Дата: | 2016 |
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| Автори: | , , |
| Формат: | Стаття |
| Мова: | English |
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Інститут прикладної математики і механіки НАН України
2016
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| Назва видання: | Український математичний вісник |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/140893 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | 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. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-140893 |
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irk-123456789-1408932018-07-18T01:23:44Z On a model semilinear elliptic equation in the plane Gutlyanskii, V.Y. Nesmelova, O.V. Ryazanov, V.I. 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). 2016 Article 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 http://dspace.nbuv.gov.ua/handle/123456789/140893 en Український математичний вісник Інститут прикладної математики і механіки НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| 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). |
| format |
Article |
| author |
Gutlyanskii, V.Y. Nesmelova, O.V. Ryazanov, V.I. |
| spellingShingle |
Gutlyanskii, V.Y. Nesmelova, O.V. Ryazanov, V.I. On a model semilinear elliptic equation in the plane Український математичний вісник |
| author_facet |
Gutlyanskii, V.Y. Nesmelova, O.V. Ryazanov, V.I. |
| author_sort |
Gutlyanskii, V.Y. |
| title |
On a model semilinear elliptic equation in the plane |
| 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 |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2016 |
| url |
http://dspace.nbuv.gov.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. |
| series |
Український математичний вісник |
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AT gutlyanskiivy onamodelsemilinearellipticequationintheplane AT nesmelovaov onamodelsemilinearellipticequationintheplane AT ryazanovvi onamodelsemilinearellipticequationintheplane |
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2023-10-18T21:23:41Z |
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2023-10-18T21:23:41Z |
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