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arXiv · 2602.15621

Completeness theorems on the boundary for a parabolic equation

Alberto Cialdea·Carmine Sebastiano Mare
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Abstract

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

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Alberto Cialdea, Carmine Sebastiano Mare. 2026-02-17. Completeness theorems on the boundary for a parabolic equation. https://arxiv.org/abs/2602.15621

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