Solvability of Planar Complex Vector Fields with Applications to Deformation of Surfaces

  • Abdelhamid Meziani
Conference paper
Part of the Trends in Mathematics book series (TM)

Abstract

Properties of solutions of a class of semilinear equations of the form Lu = f(x, y, u), where L is a ℂ-valued planar vector field is studied. Series and integral representations are obtained in a tubular neighborhood of the characteristic curve of L. An application to infinitesimal bendings of surfaces with nonnegative curvature in ℝ3 is given.

Keywords

CR equation normalization asymptotic directions infinitesimal bending 

Mathematics Subject Classification (2000)

Primary 35F05 53A05 Secondary 30G20 35C15 
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References

  1. [1]
    M.S. Baouendi and F. Treves: A property of the functions and distributions annihilated by a locally integrable system of complex vector fields, Ann. of Math., Vol. 113, 387–421, (1981)CrossRefMathSciNetGoogle Scholar
  2. [2]
    M.S. Baouendi and F. Treves: A local constancy principle for the solutions of certain overdetermined systems of first-order linear partial differential equations, Math. Analysis and Applications, Part A, Adv. in Math., Supplementary Studies, Vol. 7A, 245–262, (1981)MathSciNetGoogle Scholar
  3. [3]
    A.P. Bergamasco, P. Cordaro and J. Hounie: Global properties of a class of vector fields in the plane, J. Diff. Equations, Vol. 74, 179–199, (1988)MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    A.P. Bergamasco and A. Meziani: Solvability near the characteristic set for a class of planar vector fields of infinite type, Ann. Inst. Fourier, Vol. 55, 77–112, (2005)MATHMathSciNetGoogle Scholar
  5. [5]
    S. Berhanu and J. Hounie: Uniqueness for locally integrable solutions of overdetermined systems, Duke Math. J., Vol. 105, 387–410, (2000)MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    S. Berhanu, J. Hounie and P. Santiago: A generalized similarity principle for complex vector fields and applications, Trans. Amer. Math. Soc., Vol. 353, 1661–1675, (2001)MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    S. Berhanu and A. Meziani: Global properties of a class of planar vector fields of infinite type, Comm. PDE, Vol. 22, 99–142, (1997)MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    P. Cordaro and X. Gong: Normalization of complex-valued vector fields which degenerate along a real curve, Adv. Math., Vol. 184, 89–118, (2004)MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    J. Hounie and P. Santiago: On the local solvability of semilinear equations, Comm. PDE, Vol. 20, 1777–1789, (1995)MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    A. Meziani: On the similarity principle for planar vector fields: applications to second-order pde, J. Diff. Equations, Vol. 157, 1–19, (1999)MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    A. Meziani: On planar elliptic structures with infinite type degeneracy, J. Funct. Anal., Vol. 179, 333–373, (2001)MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    A. Meziani: Generalized CR equation with a punctual singularity: the model case, Complex Variables, Vol.48 (6), 495–512, (2003)MATHMathSciNetGoogle Scholar
  13. [13]
    A. Meziani: Generalized CR equation with a singularity, Complex Variables, Vol. 48(9), 739–752, (2003)MATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    A. Meziani: Infinitesimal bending of homogeneous surfaces with nonnegative curvature, Comm. Anal. Geom., Vol. 11, 697–719, (2003)MATHMathSciNetGoogle Scholar
  15. [15]
    A. Meziani: Elliptic planar vector fields with degeneracies, Trans. of the Amer. Math. Soc., Vol. 357, 4225–4248, (2004)CrossRefMathSciNetGoogle Scholar
  16. [16]
    A. Meziani: Planar complex vector fields and infinitesimal bendings of surfaces with nonnegative curvature, in: Contemp. Math., Vol. 400, Amer. Math. Soc., 189–202, (2006)MathSciNetGoogle Scholar
  17. [17]
    A. Meziani: Infinitesimal bendings of high orders for homogeneous surfaces with positive curvature and a flat point, J. Diff. Equations, Vol. 239, 16–37, (2007)MATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    A. Meziani: Representation of solutions of a singular CR equation in the plane, Complex Variables and Elliptic Equations, Vol. 53 (12), 1111–1130, (2008)MATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    I.K. Sabitov: Local Theory of Bending of Surfaces, Encyclopaedia Math. Sci., Vol. 48, Springer-Verlag, (1992)Google Scholar
  20. [20]
    M Spivak: A Comprehensive Introduction to Differential Geometry, Vol. 5, Publish or Perish, Berkeley, (1975)Google Scholar
  21. [21]
    F. Treves: Remarks about certain first-order linear PDE in two variables, Comm. PDE, Vol. 5, 381–425, (1980)CrossRefMathSciNetGoogle Scholar
  22. [22]
    F. Treves: Hypo-analytic structures, Princeton University Press, (1992)Google Scholar
  23. [23]
    Z.D. Usmanov: Generalized Cauchy-Riemann system with a singular point, Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 85, Longman Harlow, (1997)Google Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  • Abdelhamid Meziani
    • 1
  1. 1.Department of MathematicsFlorida International UniversityMiamiUSA

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