On a model semilinear elliptic equation in the plane
Vernadsky National Library of Ukraine
Переглянути архів Інформація| Поле | Співвідношення | |
| Title |
On a model semilinear elliptic equation in the plane
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| Creator |
Gutlyanskii, V.Y.
Nesmelova, O.V. Ryazanov, V.I. |
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| 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).
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| Date |
2018-07-17T17:51:44Z
2018-07-17T17:51:44Z 2016 |
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| Type |
Article
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| Identifier |
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 |
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| Language |
en
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| Relation |
Український математичний вісник
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| Publisher |
Інститут прикладної математики і механіки НАН України
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