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Solvability near the characteristic set for a class of planar vector fields of infinite type. (English) Zbl 1063.35051
Summary: We study the solvability of equations associated with a complex vector field $$L$$ in $${\mathbb R}^2$$ with $$C^\infty$$ or $$C^\omega$$ coefficients. We assume that $$L$$ is elliptic everywhere except on a simple and closed curve $$\Sigma$$. We assume that, on $$\Sigma$$, $$L$$ is of infinite type and that $$L\wedge\overline{L}$$ vanishes to a constant order. The equations considered are of the form $$Lu=pu+f$$, with $$f$$ satisfying compatibility conditions. We prove, in particular, that when the order of vanishing of $$L\wedge\overline{L}$$ is $$>1$$, the equation $$Lu=f$$ is solvable in the $$C^\infty$$ category but not in the $$C^\omega$$ category.

##### MSC:
 35F05 Linear first-order PDEs 30G20 Generalizations of Bers and Vekua type (pseudoanalytic, $$p$$-analytic, etc.)
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