Completeness theorems on the boundary for a parabolic equation

TL;DR

This paper establishes the completeness of polynomial solutions for a class of parabolic equations on the boundary of a domain, extending to adjoint equations, with implications for boundary value problems.

Contribution

It proves completeness theorems for polynomial solutions of parabolic equations and their adjoints on the boundary of bounded domains with smooth boundaries.

Findings

Polynomial solutions form a complete system in boundary $L^p$ spaces.

Results apply to both the original and adjoint parabolic equations.

The theorems hold for equations with constant coefficient elliptic operators.

Abstract

Let $\{v_{\alpha}\}$ 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 $\Omega_{T}$ 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 $\{v_{\alpha}\}$ is complete in $L^{p}(\Sigma')$, where $\Sigma'$ is the parabolic boundary of $\Omega_{T}$. Similar results are proved for the adjoint equation $a_{hk}\partial_{x_{h}x_{k}} u+ \partial_t u =0$.