On the similarity principle for planar vector fields: Application to second order PDE
type: Item
instance of: scholarly article
title: On the similarity principle for planar vector fields: Application to second order PDE
zbMATH Open document ID: 0937.35079
author: Abdelhamid Meziani
published in: Journal of Differential Equations
publication date: 2000-06-07T00:00:00Z
review text: The author is mainly interested in the local solvability of the equation (1) \(Lw=Aw=B\overline w\) where \(L\) is a nonsingular, integrable, \(\mathbb{C}\)-valued vector field in the plane, and \(A\) and \(B\) are smooth functions. As an application, the solutions of the second order partial differential equation (2) \(a\varphi_{xx}+ 2b\varphi_{xy} +c\varphi_{yy} +\alpha\varphi_x + \delta \varphi_y= 0\) which is elliptic everywhere except along the curve \(\Sigma= \{(x,y)\mid \rho^2(x,y)= b^2(x,y)-a(x,y) \cdot c(x,y)=0\}\) for which \(\Sigma\) is a characteristic curve, are expressed in terms of the solutions of an equation of type (1). To this end the author investigates in the paper an extension to which similarity principle holds for nonelliptic vector fields. As a paradigm to vector fields the author considers three typical vector fields: \(L_1 ={\partial \over \partial y}-3iy^2{\partial\over\partial x}\), \(L_2= {\partial \over\partial y}-i x{\partial\over \partial x}\), and \(M={\partial\over \partial y}-2iy{\partial \over\partial x}\).
Mathematics Subject Classification ID: 35J70
Mathematics Subject Classification ID: 35Q15
Mathematics Subject Classification ID: 30G20
zbMATH DE Number: 1346389
zbMATH Keywords: Vekua systems
zbMATH Keywords: similarity principle
zbMATH Keywords: nonelliptic vector field
zbMATH Keywords: characteristic
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
full work available at URL: jdeq.1999.3627
OpenAlex ID: W2050984578
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DOI: 10.1006/JDEQ.1999.3627
description: scientific article; zbMATH DE number 1346389