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Semiglobal solvability of a class of planar vector fields of infinite type. (English) Zbl 0983.35036
The authors consider a special class of planar complex-valued vector fields $$L$$ having the unit circle $$\Sigma\subset \mathbb{R}\times S^1$$, $$\Sigma= \{0\}\times S^1$$ as characteristic set. The vector field $$L$$ is of infinite type along $$\Sigma$$ and satisfies the famous Nirenberg-Treves condition for local solvability. Theorem 2.1 deals with a necessary and sufficient condition for analytic solvability of the equation $$Lu=f$$ near $$\Sigma$$, while in Theorem 3.1 it is shown that in general $$Lu= f\in C^\infty$$ does not have $$C^\infty$$ solutions in any neighbourhood of $$\Sigma$$.

##### MSC:
 35F05 Linear first-order PDEs 35A05 General existence and uniqueness theorems (PDE) (MSC2000)