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 P _m 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.