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

P. Bergamasco, Alberto; Meziani, Abdelhamid

doi:10.5802/aif.2090

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]

P. Bergamasco, Alberto ; Meziani, Abdelhamid

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

Résumé
Abstract

On étudie la résolubilité des équations associées à un champ de vecteurs complexe $L$ dans $ℝ^{2}$ à coefficients de classe $C^{\infty}$ ou $C^{ω}$ . 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 $L \land \bar{L}$ s’annule à un ordre constant. Les équations considerées sont de la forme $L u = p u + f$ , où $f$ satisfait des conditions de compatibilité. On prouve, en particulier, que lorsque l’ordre d’annulation de $L \land \bar{L}$ est $> 1$ , l’équation $L u = f$ est résoluble dans la catégorie $C^{\infty}$ mais pas dans la catégorie $C^{ω}$ .

We study the solvability of equations associated with a complex vector field $L$ in $ℝ^{2}$ with $C^{\infty}$ 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 $L \land \bar{L}$ vanishes to a constant order. The equations considered are of the form $L u = p u + f$ , with $f$ satisfying compatibility conditions. We prove, in particular, that when the order of vanishing of $L \land \bar{L}$ is $> 1$ , the equation $L u = f$ is solvable in the $C^{\infty}$ category but not in the $C^{ω}$ category.

MR 2141289 | Zbl 1063.35051

DOI : https://doi.org/10.5802/aif.2090
Classification : 35F05, 30G20
Mots clés : ensemble caractéristique, champ de vecteur complexe, type infini, résolubilité

@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},
     pages = {77--112},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {55},
     number = {1},
     year = {2005},
     doi = {10.5802/aif.2090},
     mrnumber = {2141289},
     zbl = {1063.35051},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2090/}
}

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. https://aif.centre-mersenne.org/articles/10.5802/aif.2090/

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