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://archive.numdam.org/articles/10.5802/aif.2090/

[B1] A. Bergamasco Perturbation of globally hypoelliptic operators, J. Differential Equations, Volume 114 (1994), pp. 513-526 | DOI | MR | Zbl

[B2] A. Bergamasco Remarks about global analytic hypoellipticity, Trans. Amer. Math. Soc., Volume 351 (1999), pp. 4113-4126 | DOI | MR | Zbl

[BCH] A. Bergamasco; P. Cordaro; J. Hounie Global properties of a class of vector fields in the plane, J. Diff. Equations, Volume 74 (1988), pp. 179-199 | DOI | MR | Zbl

[BCM] A. Bergamasco; P. Cordaro; P. Malagutti Globally hypoelliptic systems of vector fields, J. Funct. Analysis, Volume 114 (1993), pp. 267-285 | DOI | MR | Zbl

[BCP] A. Bergamasco; P. Cordaro; G. Petronilho Global solvability for a class of complex vector fields on the two-torus, Comm. Partial Differential Equations, Volume 29 (2004), pp. 785-819 | DOI | MR | Zbl

[BgM] A. Bergamasco; A. Meziani Semiglobal solvability of a class of planar vector fields of infinite type, Mat. Contemp., Volume 18 (2000), pp. 31-42 | MR | Zbl

[BhM1] S. Berhanu; A. Meziani On rotationally invariant vector fields in the plane, Manuscripta Math., Volume 89 (1996), pp. 355-371 | DOI | MR | Zbl

[BhM2] S. Berhanu; A. Meziani Global properties of a class of planar vector fields of infinite type, Comm. Partial Differential Equations, Volume 22 (1997), pp. 99-142 | MR | Zbl

[BHS] S. Berhanu; G. Hounie; P. Santiago A generalized similarity principle for complex vector fields and applications, Trans. Amer. Math. Soc., Volume 353 (2001), pp. 1661-1675 | DOI | MR | Zbl

[BT] M. S. Baouendi; F. Treves A property of functions and distributions annihilated by locally integrable system of vector fields, Ann. of Math., Volume 113 (1981), pp. 341-421 | MR | Zbl

[CG] P. Cordaro; X. Gong Normalization of complex-valued planar vector fields which degenerate along a real curve, Adv. Math., Volume 184 (2004), pp. 89-118 | DOI | MR | Zbl

[CH] P. Cordaro; A. Himonas Global analytic hypoellipticity for a class of degenerate elliptic operators on the torus, Math. Res. Letters, Volume 1 (1994), pp. 501-510 | MR | Zbl

[CT] P. Cordaro; F. Treves Homology and cohomology in hypoanalytic structures of the hypersurface type, J. Geo. Analysis, Volume 1 (1991), pp. 39-70 | MR | Zbl

[G] S. Greenfield Hypoelliptic vector fields and continued fractions, Proc. Amer. Math. Soc., Volume 31 (1972), pp. 115-118 | DOI | MR | Zbl

[GPY] T. Gramchev; P. Popivanov; M. Yoshino Global properties in spaces of generalized functions on the torus for second-order differential operators with variable coefficients, Rend. Sem. Mat. Univ. Pol. Torino, Volume 51 (1993), pp. 145-172 | MR | Zbl

[H] L. Hörmander The analysis of linear partial differential operators IV, New York, 1984 | MR | Zbl

[M1] A. Meziani On the similarity principle for planar vector fields: applications to second order pde, J. Differential Equations, Volume 157 (1999), pp. 1-19 | DOI | MR | Zbl

[M2] A. Meziani On real analytic planar vector fields near the characteristic set, Contemp. Math., Volume 251 (2000), pp. 429-438 | MR | Zbl

[M3] A. Meziani On planar elliptic structures with infinite type degeneracy, J. Funct. Anal., Volume 179 (2001), pp. 333-373 | DOI | MR | Zbl

[M4] A. Meziani Elliptic planar vector fields with degeneracies (Trans. Amer. Math. Soc., to appear) | MR | Zbl

[NT] L. Nirenberg; F. Treves Solvability of a first order linear partial differential equation, Comm. Pure Applied Math., Volume 16 (1963), pp. 331-351 | DOI | MR | Zbl

[T1] F. Treves Remarks about certain first-order linear PDE in two variables, Comm. Partial Differential Equations, Volume 5 (1980), pp. 381-425 | DOI | MR | Zbl

[T2] F. Treves Hypo-analytic structures: local theory, Princeton University Press, 1992 | MR | Zbl

[V] I. Vekua Generalized analytic functions, 1962 | MR | Zbl

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