Osaka Journal of Mathematics

Second order type-changing equations for a scalar function on a plane

Takahiro Noda and Kazuhiro Shibuya

Full-text: Open access


In this paper, we consider type-changing equations for one unknown function of two variables by using the theory of differential systems. We give fundamental properties and provide a notion of geometric solutions from a viewpoint of contact geometry of second order. Moreover, we study the structure of associated overdetermined systems and obtain an existence condition of solutions of a special class which are called parabolic solutions of type-changing equations.

Article information

Osaka J. Math., Volume 49, Number 1 (2012), 101-124.

First available in Project Euclid: 21 March 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58A15: Exterior differential systems (Cartan theory)
Secondary: 58A17: Pfaffian systems


Noda, Takahiro; Shibuya, Kazuhiro. Second order type-changing equations for a scalar function on a plane. Osaka J. Math. 49 (2012), no. 1, 101--124.

Export citation


  • R.L. Bryant, S.S. Chern, R. Gardner, H. Goldscmidt and P. Griffiths: Exterior Differential Systems, Mathematical Sciences Research Institute Publications 18, Springer, New York, 1991.
  • E. Cartan: Les systèmes de Pfaff, à cinq variables et les équations aux dérivées partielles du second ordre, Ann. Sci. École Norm. Sup. (3) 27 (1910), 109–192.
  • J. Clelland, M. Kossowski and G.R. Wilkens: Second-order type-changing evolution equations with first-order intermediate equations, J. Differential Equations 244 (2008), 242–273.
  • A. Hayakawa, G. Ishikawa, S. Izumiya and K. Yamaguchi: Classification of generic integral diagrams and first order ordinary differential equations, Internat. J. Math. 5 (1994), 447–489.
  • T.A. Ivey and J.M. Landsberg: Cartan for Beginners, Amer. Math. Soc., Providence, RI, 2003.
  • T. Noda and K. Shibuya, On implicit second order PDE of a scalar function on a plane, submitted.
  • H. Sato: Contact geometry of second order partial differential equations: from Darboux and Goursat, through Cartan, to modern mathematics, Sugaku Expositions 20 (2007), 137–148.
  • K. Shibuya: On the prolongation of 2-jet space of 2 independent and 1 dependent variables, Hokkaido Math. J. 38 (2009), 587–626.
  • N. Tanaka: On generalized graded Lie algebras and geometric structures I, J. Math. Soc. Japan 19 (1967), 215–254.
  • N. Tanaka: On the equivalence problems associated with simple graded Lie algebras, Hokkaido Math. J. 8 (1979), 23–84.
  • K. Yamaguchi: Contact geometry of higher order, Japan. J. Math. (N.S.) 8 (1982), 109–176.
  • K. Yamaguchi: Geometrization of jet bundles, Hokkaido Math. J. 12 (1983), 27–40.
  • K. Yamaguchi: Differential systems associated with simple graded Lie algebras; in Progress in Differential Geometry, Adv. Stud. Pure Math. 22, Math. Soc. Japan, Tokyo, 1993, 413–494.
  • K. Yamaguchi: Contact geometry of second order I; in Differential Equations: Geometry, Symmetries and Integrability, Abel Symp. 5, Springer, Berlin, 2009, 335–386.
  • K. Yamaguchi: $G_{2}$-geometry of overdetermined systems of second order; in Analysis and Geometry in Several Complex Variables (Katata, 1997), Trends Math, Birkhäuser, Boston, MA, 1999, 289–314.