# zbMATH — the first resource for mathematics

Classes of globally solvable involutive systems. (English) Zbl 1382.58018
Let $$M$$ be a smooth closed orientable surface and $$d$$ the de Rham operator on it. Let $$\mathbb{S}^1$$ be the unit circle with the standard unit vector field $$\partial$$. Let $$b$$ be a closed non-exact 1-form on $$M$$. Consider the following complex $0\to \mathcal{D}'(M\times\mathbb{S}^1)\overset{\mathbb{L}}{\to} \bigwedge{}^1\,\mathcal{D}'(M\times\mathbb{S}^1)\overset{\mathbb{L}^1}{\to} \bigwedge{}^2\,\mathcal{D}'(M\times\mathbb{S}{^1)} 0,$ where both operators $$\mathbb{L}$$ and $$\mathbb{L}^1$$ are defined by the formula $\mathbb{L}u=du+ib\wedge\partial u.$ The operator $$\mathbb{L}$$ defines a first order overdetermined system of linear partial differential equations, whose (local) compatibility conditions are given by the operator $$\mathbb{L}^1$$.
The main theorem states that the global solvability of $$\mathbb{L}$$ is equivalent to the condition that no super- or sub-level of a potential $$B$$ of the pull-back of $$b$$ to the universal cover of $$M$$ has a bounded component. Moreover, on a certain covering space $$\tilde{M}$$, called minimal covering space, the condition is equivalent to the connectedness of the superlevel and sublevel sets of the corresponding pseudo-periodic potential $$\tilde{B}$$.
Some versions of this problem have been investigated before in the works of F. Tréves, F. Cardoso, J. Hounie and others.

##### MSC:
 58J10 Differential complexes 35A01 Existence problems for PDEs: global existence, local existence, non-existence 35N10 Overdetermined systems of PDEs with variable coefficients
Full Text:
##### References:
 [1] Arnol’d, VI, Topological and ergodic properties of closed 1-forms with incommensurable periods, Funkt. Anal. Prilozhen., 25, 1-12, (1991) · Zbl 0732.58001 [2] Bergamasco, AP; Kirilov, A, Global solvability for a class of overdetermined systems, J. Funct. Anal., 252, 603-629, (2007) · Zbl 1158.58011 [3] Bergamasco, AP; Zani, SL, Global analytic regularity for structures of co-rank one, Commun. Partial Differ. Equ., 33, 933-941, (2008) · Zbl 1153.35006 [4] Bergamasco, AP; Cordaro, PD; Malagutti, PA, Globally hypoelliptic systems of vector fields, J. Funct. Anal., 114, 267-285, (1993) · Zbl 0777.58041 [5] Bergamasco, AP; Cordaro, PD; Petronilho, G, Global solvability for certain classes of underdetermined systems of vector fields, Math. Z., 223, 261-274, (1996) · Zbl 0863.58062 [6] Bergamasco, AP; Kirilov, A; Nunes, WVL; Zani, SL, On the global solvability for overdetermined systems, Trans. Am. Math. Soc., 364, 4533-4549, (2012) · Zbl 1275.35004 [7] Bergamasco, AP; Medeira, C; Zani, SL, Globally solvable systems of complex vector fields, J. Differ. Equ., 252, 4598-4623, (2012) · Zbl 1242.35092 [8] Bergamasco, AP; Nunes, WVL; Zani, SL, Global properties of a class of overdetermined systems, J. Funct. Anal., 200, 31-64, (2003) · Zbl 1034.32024 [9] Berhanu, S., Cordaro, P.D., Hounie, J.: An Introduction to Involutive Structures. New Mathematical Monographs, vol. 6. Cambridge University Press, Cambridge (2008). doi:10.1017/CBO9780511543067 · Zbl 1151.35011 [10] Cardoso, F; Hounie, J, Global solvability of an abstract complex, Proc. Am. Math. Soc., 65, 117-124, (1977) · Zbl 0335.58015 [11] Farber, M.: Topology of Closed One-Forms. Mathematical Surveys and Monographs, vol. 108. American Mathematical Society, Providence (2004). doi:10.1090/surv/108 · Zbl 1052.58016 [12] Fenn, R, What is the geometry of a surface?, Am. Math. Mon., 90, 87-98, (1983) · Zbl 0528.57006 [13] Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002) · Zbl 1044.55001 [14] Hounie, J., Zugliani, G.: Global solvability of real analytic involutive systems on compact manifolds. Math. Ann. 1-33 (2016). doi:10.1007/s00208-016-1471-5 · Zbl 1380.35129 [15] Jost, J.: Compact Riemann Surfaces, An Introduction to Contemporary Mathematics. Universitext, 3rd edn. Springer, Berlin (2006). doi:10.1007/978-3-540-33067-7 · Zbl 1125.30033 [16] Palais, RS, Natural operations on differential forms, Trans. Am. Math. Soc., 92, 125-141, (1959) · Zbl 0092.30802 [17] Trèves, F.: Topological Vector Spaces, Distributions and Kernels. Academic Press, New York (1967) · Zbl 0171.10402 [18] Treves, F, Study of a model in the theory of complexes of pseudodifferential operators, Ann. Math., 104, 269-324, (1976) · Zbl 0354.35067 [19] Trèves, F.: Hypo-Analytic Structures, Local Theory. Princeton Mathematical Series, vol. 40. Princeton University Press, Princeton (1992) · Zbl 0787.35003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.