Completeness theorems on the boundary for a parabolic equation

Abstract

Let $\{v_{\hat{I}}\pm \}$ be a system of polynomial solutions of the parabolic equation $a_{hk}{\partial }_{x_h{x}_k}u-{\partial }_{t}u=0$ in a bounded $C^1$-cylinder $\hat{I}{\text{c◯}}_T$ contained in ${\mathbb{R}}^{n+1}$. Here $a_{hk}{\partial }_{x_h{x}_k}$ is an elliptic operator with real constant coefficients. We prove that $\{v_{\hat{I}}\pm \}$ is complete in $L^p(\hat{I}{\pounds }^{\prime })$, where $\hat{I}{\pounds }^{\prime }$ is the parabolic boundary of $\hat{I}{\text{c◯}}_T$. Similar results are proved for the adjoint equation $a_{hk}{\partial }_{x_h{x}_k}u+{\partial }_{t}u=0$.

24 pages