Complete prolongation and the Frobenius integrability for overdetermined systems of partial differential equations

Abstract

We study the compatibility conditions and the existenceof solutions for overdetermined PDE systems that admitcomplete prolongation. For a complete system of order kthere exists a submanifold of the (k-1)st jet space of unknown functionsthat is the largest possible set on which the initialconditions of (k-1)st order may take values. Thereexists a unique solution for any initial condition thatbelongs to this set if and only if the complete systemsatisfies the compatibility conditions on the initialdata set. We prove by applying the Frobenius theorem toa Pfaffian differential system associated with the completeprolongation.