Completeness Theorems for Elliptic Equations of Higher Order with Constant Coefficients

Let $\{\omega_{k}\}$ be a complete system of polynomial solutions of the elliptic equation $\sum_{|\alpha|\leq 2m}a_{\alpha}D^{\alpha}u=0$, $a_{\alpha}$ being real constants. We give necessary and sufficient conditions for the completeness of the system $\{(\omega_{k},\de_{\nu}\omega_{k},\ldots,\de_{\nu}^{m-1}\omega_{k})\}$ in $[L^{p}(\partial\Omega)]^{m}$, where $\Omega\subset R^{n}$ is a bounded domain such that $R^{n}\setminus\overline{\Omega}$ is connected and $\partial\Omega\in C^{1}$