Linear Overdetermined Systems of Partial Differential Equations. Initial and Initial-Boundary Value Problems

Consider a linear partial differential operator A that maps a vector-valued function y= (y ₁ , ...,y _{m )} into a vector-valued function f = (f ₁ ,...,f _l ) We assume at first that all the functions, as well as the coefficients of the differential operator, are defined in an open domain Ω in the n-dimensional Euclidean space ℝn, and that they are smooth (infinitely differentiable). A is called an overdetermined operator if there is a non-zero differential operator A′ such that the composition A′A is the zero operator (and underdetermined if there is a non-zero operator A″ such that AA″ = 0). If A is overdetermined, then A′ f = 0 is a necessary condition for the solvability of the system A _y = f with an unknown vector-valued function y.