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, p. 77-112
We study the solvability of equations associated with a complex vector field L in ℝ 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 L∧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∧L ¯ is >1, the equation Lu=f is solvable in the C ∞ category but not in the C ω category.
On étudie la résolubilité des équations associées à un champ de vecteurs complexe L dans ℝ 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 L∧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∧L ¯ est >1, l’équation Lu=f est résoluble dans la catégorie C ∞ mais pas dans la catégorie C ω .
DOI : https://doi.org/10.5802/aif.2090
Classification:  35F05,  30G20
Keywords: characteristic set, complex vector field, infinite type, solvability 
@article{AIF_2005__55_1_77_0,
     author = {P. Bergamasco, Alberto and Meziani, Abdelhamid},
     title = {Solvability near the characteristic set for a class of planar vector fields of infinite type},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {55},
     number = {1},
     year = {2005},
     pages = {77-112},
     doi = {10.5802/aif.2090},
     mrnumber = {2141289},
     zbl = {1063.35051},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2005__55_1_77_0}
}

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/item/AIF_2005__55_1_77_0/

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