On the extension of solutions of linear constant-coefficient PDE

Abstract

We solve the following over-determined boundary value problem (the “extension problem”): 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 g, we give a criterion for the (unique) existence of a smooth solution f to this problem. There are two obvious necessary conditions: g is smooth and 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 conditions. 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.