Solvability near the characteristic set for a class of planar vector fields of infinite type.

*(English)*Zbl 1063.35051Summary: 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.) |

##### Keywords:

complex vector field; \(C^\infty\) coefficients; \(C^\omega\) coefficients; compatibility conditions##### References:

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