Global properties of a class of vector fields in the plane. (English) Zbl 0662.58021

Bergamasco, A.; Cordaro, P.; Hounie, J.

J. Differ. Equations 74, No. 2, 179-199 (1988).

The paper is concerned with the global solvability of the problem \(Ln=0\), dn\(\neq 0\) on \({\mathbb{R}}^ 2\), where L is a complex vector field without singularities. First it is shown that for a suitable class of vector fields L the Mizohata operator \(\partial_ t-it\partial y\) is a model operator in a neighborhood of the characteristic set of L. Then several integrability conditions are discussed and some global range theorems for the Mizohata operator are given. In an appendix relations to hyperelliptic vector fields are considered.

Reviewer: N.Jacob

MSC:

37J35	Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10	Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)

Keywords:

vector fields in the plane; Mizohata operator; integrability conditions

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