Solvability of systems of partial differential equations for functions defined on nonconvex sets

Abstract

We consider systems of partial differential equations with constant coefficients of the form \(\big ( R(D_x, D_y)f = 0, P(D_x)f = {g}\big ), f,g \in {C}^{\infty}(\Omega),\), where R (and P) are operators in (n + 1) variables (and in n variables, respectively), g satisfies the compatibility condition \(R(D_x, D_y){g} = 0 \ {\rm and} \ \Omega \subset {\Bbb R}^{n+1}\) is open. Let R be elliptic. We show that the solvability of such systems for certain nonconvex sets \(\Omega \) implies that any localization at \(\infty \) of the principle part Pm of P is hyperbolic. In contrast to this result such systems can always be solved on convex open sets \(\Omega \) by the fundamental principle of Ehrenpreis-Palamodov.