Global properties of a class of overdetermined systems.
type: Item
instance of: scholarly article
title: Global properties of a class of overdetermined systems.
zbMATH Open document ID: 1034.32024
DOI: 10.1016/S0022-1236(02)00055-1
author: Wagner Vieira Leite Nunes
author: Sérgio Luís Zani
author: Adalberto P. Bergamasco
published in: Journal of Functional Analysis
publication date: 2003-05-25T00:00:00Z
review text: The authors study a class of overdetermined systems of two complex linear differential equations on the three-dimensional real torus with one unknown function described as follows. Let \(\varphi=a\,dt + b\,ds\) be a real analytic closed differential \(1\)-form on the two-dimensional torus with periodic coordinates \((t,s)\) and let \(\Pi\) be the product of the torus and the circle with coordinate \(x\). Then \(\psi=dx + i\varphi, \, i = \sqrt{-1},\) is a complex \(1\)-form on \(\Pi.\) The operator of the system in question is formed by a pair of complex vector fields \(\upsilon_1=\frac{\partial}{\partial t} - i a\frac{\partial}{\partial x }\) and \(\upsilon_2=\frac{\partial}{\partial s} - i b\frac{\partial}{\partial x}\) which are orthogonal to \(\psi.\) In other words: the total differential \(du = u_xdx + u_sds + u_tdt\) of the unknown function is equal modulo of the constructed form \(\psi\) to \(\chi= fdt + gds\) determining the right side of the system. Problems of the existence of solutions for such systems and their smoothness are investigated in detail. In particular, it is proved that under natural requirements on compatibility and periodicity a necessary and sufficient condition of the global solvability of the system is in fact its local solvability. In a similar case the latter condition was considered by \textit{L. Nirenberg} and \textit{F. Trèves} [Commun. Pure Appl. Math. 16, 331--351 (1963; Zbl 0117.06104)]; in the context of the paper under review one can formulate it in such a way: the local primitives of the differential form \(\varphi\) determine maps which are open at all the regular points of the zero-set defined by the coefficients of the \(1\)-form. If the local condition fails then the system turns out to be non hypoelliptic. In order to prove this claim the authors construct a non-smooth function which is transformed into smooth ones by the vector fields \(\upsilon_1\) and \(\upsilon_2.\)
Mathematics Subject Classification ID: 32W50
Mathematics Subject Classification ID: 35N10
Mathematics Subject Classification ID: 35H10
zbMATH DE Number: 1915619
zbMATH Keywords: global solvability
zbMATH Keywords: hypoelliptic systems
zbMATH Keywords: analytic singularities
zbMATH Keywords: level sets
zbMATH Keywords: pseudoperiodic functions
zbMATH Keywords: commensurable periods
zbMATH Keywords: locally integrable structures
zbMATH Keywords: Euler graph
zbMATH Keywords: stationary phase
zbMATH Keywords: overdetermined systems
zbMATH Keywords: three-dimensional real torus
reviewed by: Aleksandr G. Aleksandrov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1915619