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},
     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 = {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, Tome 55 (2005) no. 1, pp. 77-112. doi : 10.5802/aif.2090. https://aif.centre-mersenne.org/item/AIF_2005__55_1_77_0/

Bibliographie

[B1] A. Bergamasco Perturbation of globally hypoelliptic operators, J. Differential Equations, Tome 114 (1994), pp. 513-526 | Article | MR 1303037 | Zbl 0815.35009

[B2] A. Bergamasco Remarks about global analytic hypoellipticity, Trans. Amer. Math. Soc., Tome 351 (1999), pp. 4113-4126 | Article | MR 1603878 | Zbl 0932.35046

[BCH] A. Bergamasco; P. Cordaro; J. Hounie Global properties of a class of vector fields in the plane, J. Diff. Equations, Tome 74 (1988), pp. 179-199 | Article | MR 952894 | Zbl 0662.58021

[BCM] A. Bergamasco; P. Cordaro; P. Malagutti Globally hypoelliptic systems of vector fields, J. Funct. Analysis, Tome 114 (1993), pp. 267-285 | Article | MR 1223704 | Zbl 0777.58041

[BCP] A. Bergamasco; P. Cordaro; G. Petronilho Global solvability for a class of complex vector fields on the two-torus, Comm. Partial Differential Equations, Tome 29 (2004), pp. 785-819 | Article | MR 2059148 | Zbl 1065.35088

[BgM] A. Bergamasco; A. Meziani Semiglobal solvability of a class of planar vector fields of infinite type, Mat. Contemp., Tome 18 (2000), pp. 31-42 | MR 1812861 | Zbl 0983.35036

[BhM1] S. Berhanu; A. Meziani On rotationally invariant vector fields in the plane, Manuscripta Math., Tome 89 (1996), pp. 355-371 | Article | MR 1378599 | Zbl 0858.35021

[BhM2] S. Berhanu; A. Meziani Global properties of a class of planar vector fields of infinite type, Comm. Partial Differential Equations, Tome 22 (1997), pp. 99-142 | MR 1434140 | Zbl 0882.35029

[BHS] S. Berhanu; G. Hounie; P. Santiago A generalized similarity principle for complex vector fields and applications, Trans. Amer. Math. Soc., Tome 353 (2001), pp. 1661-1675 | Article | MR 1806725 | Zbl 0982.35030

[BT] M. S. Baouendi; F. Treves A property of functions and distributions annihilated by locally integrable system of vector fields, Ann. of Math., Tome 113 (1981), pp. 341-421 | MR 607899 | Zbl 0491.35036

[CG] P. Cordaro; X. Gong Normalization of complex-valued planar vector fields which degenerate along a real curve, Adv. Math., Tome 184 (2004), pp. 89-118 | Article | MR 2047850 | Zbl 02081957

[CH] P. Cordaro; A. Himonas Global analytic hypoellipticity for a class of degenerate elliptic operators on the torus, Math. Res. Letters, Tome 1 (1994), pp. 501-510 | MR 1302393 | Zbl 0836.35036

[CT] P. Cordaro; F. Treves Homology and cohomology in hypoanalytic structures of the hypersurface type, J. Geo. Analysis, Tome 1 (1991), pp. 39-70 | MR 1097935 | Zbl 0724.32009

[G] S. Greenfield Hypoelliptic vector fields and continued fractions, Proc. Amer. Math. Soc., Tome 31 (1972), pp. 115-118 | Article | MR 301459 | Zbl 0229.35024

[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, Tome 51 (1993), pp. 145-172 | MR 1289385 | Zbl 0824.35027

[H] L. Hörmander The analysis of linear partial differential operators IV, New York, 1984 | MR 781537 | Zbl 0612.35001

[M4] A. Meziani Elliptic planar vector fields with degeneracies (Trans. Amer. Math. Soc., to appear) | Article | MR 1710011 | Zbl 0937.35079

[M1] A. Meziani On the similarity principle for planar vector fields: applications to second order pde, J. Differential Equations, Tome 157 (1999), pp. 1-19 | Article | MR 1771284 | Zbl 0960.35016

[M2] A. Meziani On real analytic planar vector fields near the characteristic set, Contemp. Math., Tome 251 (2000), pp. 429-438 | Article | MR 1809114 | Zbl 0973.35083

[M3] A. Meziani On planar elliptic structures with infinite type degeneracy, J. Funct. Anal., Tome 179 (2001), pp. 333-373 | Article | MR 2159708 | Zbl 02188287

[NT] L. Nirenberg; F. Treves Solvability of a first order linear partial differential equation, Comm. Pure Applied Math., Tome 16 (1963), pp. 331-351 | Article | MR 163045 | Zbl 0117.06104

[T1] F. Treves Remarks about certain first-order linear PDE in two variables, Comm. Partial Differential Equations, Tome 5 (1980), pp. 381-425 | Article | MR 567779 | Zbl 0523.35012

[T2] F. Treves Hypo-analytic structures: local theory, Princeton University Press, 1992 | MR 1200459 | Zbl 0787.35003

[V] I. Vekua Generalized analytic functions, 1962 | MR 150320 | Zbl 0100.07603

ANNALES DE L'INSTITUT FOURIER

ARTICLES À PARAÎTRE

FEUILLETER

PRESENTATION

COMITÉ DE RÉDACTION

SOUMETTRE UN ARTICLE

ABONNEMENT