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

Alberto P. Bergamasco^{[1]}; Abdelhamid Meziani

- [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)

Annales de l’institut Fourier (2005)

- Volume: 55, Issue: 1, page 77-112
- ISSN: 0373-0956

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topP. Bergamasco, Alberto, and Meziani, Abdelhamid. "Solvability near the characteristic set for a class of planar vector fields of infinite type." Annales de l’institut Fourier 55.1 (2005): 77-112. <http://eudml.org/doc/116192>.

@article{P2005,

abstract = {We study the solvability of equations associated with a complex vector field $L$ in
$\{\mathbb \{R\}\}^2$ 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.},

affiliation = {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)},

author = {P. Bergamasco, Alberto, Meziani, Abdelhamid},

journal = {Annales de l’institut Fourier},

keywords = {characteristic set; complex vector field; infinite type; solvability; coefficients; coefficients; compatibility conditions},

language = {eng},

number = {1},

pages = {77-112},

publisher = {Association des Annales de l'Institut Fourier},

title = {Solvability near the characteristic set for a class of planar vector fields of infinite type},

url = {http://eudml.org/doc/116192},

volume = {55},

year = {2005},

}

TY - JOUR

AU - P. Bergamasco, Alberto

AU - Meziani, Abdelhamid

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

JO - Annales de l’institut Fourier

PY - 2005

PB - Association des Annales de l'Institut Fourier

VL - 55

IS - 1

SP - 77

EP - 112

AB - We study the solvability of equations associated with a complex vector field $L$ in
${\mathbb {R}}^2$ 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.

LA - eng

KW - characteristic set; complex vector field; infinite type; solvability; coefficients; coefficients; compatibility conditions

UR - http://eudml.org/doc/116192

ER -

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