Over-determined Systems

Let R(∂) be a matrix whose entries are linear partial differential operators, with constant coefficients. Let Ω be a non-empty, open, bounded, convex set. We consider the homogeneous system R(∂)f = 0 in a neighborhood of Ω, subject to the boundary condition f = g in a neighborhood of ∂Ω. For a given smooth function g, we give a criterion for the (unique) existence of a smooth solution f to this problem. There is an obvious necessary condition: R(∂)g = 0 in a neighborhood of ∂Ω. We characterize the class of differential operators R(∂) for which the problem is solvable for any g satisfying the necessary condition. A system is over-determined if and only if there exists a unique solution to the problem, for any such Ω and g. We derive an algebraic criterion for this property. Finally, in the case where the solution is non-unique, we consider the possibility of obtaining uniqueness by fixing several components of the desired solution.