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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
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