Journal of the
Korean Mathematical Society JKMS

We study the compatibility conditions and the existence of solutions for overdetermined PDE systems that admit complete prolongation. For a complete system of order $k$ there exists a submanifold of the $(k-1)$st jet space of unknown functions that is the largest possible set on which the initial conditions of $(k-1)$st order may take values. There exists a unique solution for any initial condition that belongs to this set if and only if the complete system satisfies the compatibility conditions on the initial data set. We prove by applying the Frobenius theorem to a Pfaffian differential system associated with the complete prolongation.

Keywords: prolongation, overdetermined systems, integrability

MSC numbers: 35N10, 58A17