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

Abstract

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We study the solvability of equations associated with a complex vector field L in ℝ 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 L ∧ 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 ∧ L ¯ is > 1 , the equation L u = f is solvable in the C ∞ category but not in the C ω category.

How to cite

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P. 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 $&gt;1$, 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 $&gt;1$, 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 -

References

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  1. M. S. Baouendi, F. Treves, A property of functions and distributions annihilated by locally integrable system of vector fields, Ann. of Math. 113 (1981), 341-421 Zbl0491.35036MR607899
  2. A. Bergamasco, Perturbation of globally hypoelliptic operators, J. Differential Equations 114 (1994), 513-526 Zbl0815.35009MR1303037
  3. A. Bergamasco, Remarks about global analytic hypoellipticity, Trans. Amer. Math. Soc. 351 (1999), 4113-4126 Zbl0932.35046MR1603878
  4. A. Bergamasco, P. Cordaro, J. Hounie, Global properties of a class of vector fields in the plane, J. Diff. Equations 74 (1988), 179-199 Zbl0662.58021MR952894
  5. A. Bergamasco, P. Cordaro, P. Malagutti, Globally hypoelliptic systems of vector fields, J. Funct. Analysis 114 (1993), 267-285 Zbl0777.58041MR1223704
  6. A. Bergamasco, P. Cordaro, G. Petronilho, Global solvability for a class of complex vector fields on the two-torus, Comm. Partial Differential Equations 29 (2004), 785-819 Zbl1065.35088MR2059148
  7. A. Bergamasco, A. Meziani, Semiglobal solvability of a class of planar vector fields of infinite type, Mat. Contemp. 18 (2000), 31-42 Zbl0983.35036MR1812861
  8. S. Berhanu, G. Hounie, P. Santiago, A generalized similarity principle for complex vector fields and applications, Trans. Amer. Math. Soc. 353 (2001), 1661-1675 Zbl0982.35030MR1806725
  9. S. Berhanu, A. Meziani, On rotationally invariant vector fields in the plane, Manuscripta Math. 89 (1996), 355-371 Zbl0858.35021MR1378599
  10. S. Berhanu, A. Meziani, Global properties of a class of planar vector fields of infinite type, Comm. Partial Differential Equations 22 (1997), 99-142 Zbl0882.35029MR1434140
  11. P. Cordaro, X. Gong, Normalization of complex-valued planar vector fields which degenerate along a real curve, Adv. Math. 184 (2004), 89-118 Zbl1129.35419MR2047850
  12. P. Cordaro, A. Himonas, Global analytic hypoellipticity for a class of degenerate elliptic operators on the torus, Math. Res. Letters 1 (1994), 501-510 Zbl0836.35036MR1302393
  13. P. Cordaro, F. Treves, Homology and cohomology in hypoanalytic structures of the hypersurface type, J. Geo. Analysis 1 (1991), 39-70 Zbl0724.32009MR1097935
  14. S. Greenfield, Hypoelliptic vector fields and continued fractions, Proc. Amer. Math. Soc. 31 (1972), 115-118 Zbl0229.35024MR301459
  15. 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 51 (1993), 145-172 Zbl0824.35027MR1289385
  16. L. Hörmander, The analysis of linear partial differential operators IV, (1984), New York Zbl0612.35001MR781537
  17. A. Meziani, On the similarity principle for planar vector fields: applications to second order pde, J. Differential Equations 157 (1999), 1-19 Zbl0937.35079MR1710011
  18. A. Meziani, On real analytic planar vector fields near the characteristic set, Contemp. Math. 251 (2000), 429-438 Zbl0960.35016MR1771284
  19. A. Meziani, On planar elliptic structures with infinite type degeneracy, J. Funct. Anal. 179 (2001), 333-373 Zbl0973.35083MR1809114
  20. A. Meziani, Elliptic planar vector fields with degeneracies Zbl1246.35090MR2159708
  21. L. Nirenberg, F. Treves, Solvability of a first order linear partial differential equation, Comm. Pure Applied Math. 16 (1963), 331-351 Zbl0117.06104MR163045
  22. F. Treves, Remarks about certain first-order linear PDE in two variables, Comm. Partial Differential Equations 5 (1980), 381-425 Zbl0523.35012MR567779
  23. F. Treves, Hypo-analytic structures: local theory, (1992) Zbl0787.35003MR1200459
  24. I. Vekua, Generalized analytic functions, (1962) Zbl0100.07603MR150320

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