Rank two prolongations of second-order PDE and geometric singular solutions.

*(English)*Zbl 1301.58002The authors prove that the fibers of the first prolongation of a second order hyperbolic, parabolic or elliptic (respectively) 2nd order PDE are tori, pinched tori (with one singular point) or spheres (respectively). They compute out the tangent nilpotent graded Lie algebras that arise as symbol algebras in Tanaka’s theory for any equation of each of the three classes. They give examples of singular solutions of an equation of each type, so that the solution is a smooth surface in a suitable prolongation. The authors point out that the torus fibers in the hyperbolic case were described in a paper of R. Bryant, P. Griffiths and L. Hsu [Sel. Math., New Ser. 1, No. 1, 21–112 (1995; Zbl 0853.58102)]; the reader might also enjoy reading Eendebak’s work on tori in first order hyperbolic systems for two functions of two variables.

Reviewer: Benjamin McKay (Cork)

##### MSC:

58A15 | Exterior differential systems (Cartan theory) |

58A17 | Pfaffian systems |

58J05 | Elliptic equations on manifolds, general theory |

35R01 | PDEs on manifolds |

35J15 | Second-order elliptic equations |

35K10 | Second-order parabolic equations |

35L10 | Second-order hyperbolic equations |

##### Keywords:

exterior differential systems; implicit second order PDEs; prolongations; geometric singular solutions
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\textit{T. Noda} and \textit{K. Shibuya}, Tokyo J. Math. 37, No. 1, 73--110 (2014; Zbl 1301.58002)

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##### References:

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