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 t...
Збережено в:
| Опубліковано в: : | Український математичний вісник |
|---|---|
| Дата: | 2016 |
| Автори: | , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут прикладної математики і механіки НАН України
2016
|
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/140893 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| 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| _version_ | 1862688987587543040 |
|---|---|
| author | Gutlyanskii, V.Y. Nesmelova, O.V. Ryazanov, V.I. |
| author_facet | Gutlyanskii, V.Y. Nesmelova, O.V. Ryazanov, V.I. |
| 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. |
| collection | DSpace DC |
| container_title | Український математичний вісник |
| 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).
|
| first_indexed | 2025-12-07T16:10:15Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-140893 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1810-3200 |
| language | English |
| last_indexed | 2025-12-07T16:10:15Z |
| publishDate | 2016 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | 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 |
| spellingShingle | On a model semilinear elliptic equation in the plane Gutlyanskii, V.Y. Nesmelova, O.V. Ryazanov, V.I. |
| title | 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_short | On a model semilinear elliptic equation in the plane |
| title_sort | on a model semilinear elliptic equation in the plane |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/140893 |
| work_keys_str_mv | AT gutlyanskiivy onamodelsemilinearellipticequationintheplane AT nesmelovaov onamodelsemilinearellipticequationintheplane AT ryazanovvi onamodelsemilinearellipticequationintheplane |