Solvability near the characteristic set for a class of planar vector fields of infinite type
[Résolubilité au voisingage de l'ensemble caractéristique pour une classe de champs de vecteurs planaires de type infini]
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 2 à coefficients de classe C ou C ω . On suppose que L est partout elliptique, sauf le long d’une courbe simple et fermée Σ. Sur Σ, on suppose que L est de type infini et que LL ¯ 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 LL ¯ est >1, l’équation Lu=f est résoluble dans la catégorie C mais pas dans la catégorie C ω .

We study the solvability of equations associated with a complex vector field L in 2 with C or C ω coefficients. We assume that L is elliptic everywhere except on a simple and closed curve Σ. We assume that, on Σ, L is of infinite type and that LL ¯ 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 LL ¯ is >1, the equation Lu=f is solvable in the C category but not in the C ω category.

DOI : 10.5802/aif.2090
Classification : 35F05, 30G20
Keywords: characteristic set, complex vector field, infinite type, solvability
Mot clés : ensemble caractéristique, champ de vecteur complexe, type infini, résolubilité
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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     title = {Solvability near the characteristic set for a class of planar vector fields of infinite type},
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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, Tome 55 (2005) no. 1, pp. 77-112. doi : 10.5802/aif.2090. http://www.numdam.org/articles/10.5802/aif.2090/

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