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 ${C}^{\infty }$ 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 : 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},
zbl = {1063.35051},
mrnumber = {2141289},
language = {en},
url = {http://www.numdam.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. http://www.numdam.org/item/AIF_2005__55_1_77_0/

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