| Full text | |
| Author(s): |
Hounie, Jorge
;
Zugliani, Giuliano
Total Authors: 2
|
| Document type: | Journal article |
| Source: | Journal of Differential Equations; v. 340, p. 26-pg., 2022-12-15. |
| Abstract | |
We consider smooth, nonvanishing complex (not essentially real) vector fields L =X + i Y that satisfy the Nirenberg-Treves condition (P) and are allowed to possess some closed one-dimensional orbits in the sense of Sussmann that we call clean orbits. Concerning global solvability, we deal with two different cases -depending on whether the surface is compact or not- and prove two positive results which extend and unify several known results on the subject.In our positive results, it is assumed that X perpendicular to Y vanishes of order greater than 1 on all closed one-orbits, on the other hand, we prove that global solvability cannot hold if there exists at least one closed one-orbit on which X perpendicular to Y vanishes of order 1.We also give a characterization of globally hypoelliptic complex vector fields L in terms of the topo-logical type of the equivalence classes determined by a standard equivalence relation defined on the set of non elliptic points of L. Some of the consequences are that globally hypoelliptic vector fields are globally solvable and that if a surface M carries a globally hypoelliptic vector field, it must be parallelizable.(c) 2022 Elsevier Inc. All rights reserved. (AU) | |
| FAPESP's process: | 18/14316-3 - Geometric theory of PDE and multidimensional complex analysis |
| Grantee: | Paulo Domingos Cordaro |
| Support Opportunities: | Research Projects - Thematic Grants |
| FAPESP's process: | 21/00693-2 - Topological approaches for partial differential equations |
| Grantee: | Giuliano Angelo Zugliani |
| Support Opportunities: | Regular Research Grants |