# ANNALES DE L'INSTITUT FOURIER

Solvability near the characteristic set for a class of planar vector fields of infinite type
Annales de l'Institut Fourier, Volume 55 (2005) no. 1, pp. 77-112.

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.

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 ${C}^{\omega }$.

DOI: 10.5802/aif.2090
Classification: 35F05, 30G20
Keywords: characteristic set, complex vector field, infinite type, solvability
P. Bergamasco, Alberto 1; Meziani, Abdelhamid

1 Instituto de Ciências Matemáticas e de Computaçao-USP, Departamento de Matemática, Caixa Postal 668, 13.560-970 Sao Carlos SP (Brésil), Florida International University, Department of Mathematics, Miami, FL 33199 (USA)
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P. Bergamasco, Alberto; Meziani, Abdelhamid. Solvability near the characteristic set for a class of planar vector fields of infinite type. Annales de l'Institut Fourier, Volume 55 (2005) no. 1, pp. 77-112. doi : 10.5802/aif.2090. https://aif.centre-mersenne.org/articles/10.5802/aif.2090/

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