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Rank two prolongations of second-order PDE and geometric singular solutions. (English) Zbl 1301.58002
The 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.
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
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