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On a Model Semilinear Elliptic Equation in the Plane
- Source :
- Journal of Mathematical Sciences. 220:603-614
- Publication Year :
- 2016
- Publisher :
- Springer Science and Business Media LLC, 2016.
-
Abstract
- Assume that Ω is a regular domain in the complex plane ℂ, and A(z) is a 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) = eu 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 a quasiconformal homeomorphism generated by the matrix A(z), and T is a solution of the semilinear weihted Bieberbach equation △T = m(w)e in G. Here, the weight m(w) is the Jacobian determinant of the inverse mapping ω−1(w).
- Subjects :
- Statistics and Probability
Pure mathematics
Applied Mathematics
General Mathematics
010102 general mathematics
Mathematical analysis
Regular solution
01 natural sciences
Beltrami equation
Homeomorphism
010305 fluids & plasmas
Matrix (mathematics)
symbols.namesake
Elliptic curve
0103 physical sciences
Domain (ring theory)
Jacobian matrix and determinant
symbols
0101 mathematics
Complex plane
Mathematics
Subjects
Details
- ISSN :
- 15738795 and 10723374
- Volume :
- 220
- Database :
- OpenAIRE
- Journal :
- Journal of Mathematical Sciences
- Accession number :
- edsair.doi...........c941ef7a90f85c50bb99fdfbda00128c