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

P. Bergamasco, Alberto; Meziani, Abdelhamid

Annales de l'Institut Fourier, Tome 55 (2005) no. 1, pp. 77-112.

On étudie la résolubilité des équations associées à un champ de vecteurs complexe $L$ dans ${ℝ}^2$ à coefficients de classe $C^{\infty }$ ou $C^{\omega }$. On suppose que $L$ est partout elliptique, sauf le long d’une courbe simple et fermée $\Sigma$. Sur $\Sigma$, on suppose que $L$ est de type infini et que $L\wedge \overline{L}$ s’annule à un ordre constant. Les équations considerées sont de la forme $Lu=pu+f$, où $f$ satisfait des conditions de compatibilité. On prouve, en particulier, que lorsque l’ordre d’annulation de $L\wedge \overline{L}$ est $>1$, l’équation $Lu=f$ est résoluble dans la catégorie $C^{\infty }$ mais pas dans la catégorie $C^{\omega }$.

We study the solvability of equations associated with a complex vector field $L$ in ${ℝ}^2$ 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.

Classification : 35F05, 30G20
Keywords: characteristic set, complex vector field, infinite type, solvability
Mot clés : ensemble caractéristique, champ de vecteur complexe, type infini, résolubilité

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