Zero-mean periodic Sobolev cell space

Fix $d\ge 1$, $Y=(0,1{)}^d$, and Countable Choice for the Sobolev conventions of The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$). Let $C_{\mathrm{\#}}^{\mathrm{\infty }}(Y)$ be the restrictions to $Y$ of smooth ${\mathbb{Z}}^d$-periodic real functions on ${\mathbb{R}}^d$. Define $H_{\mathrm{\#}}^1(Y):={\overline{C_{\mathrm{\#}}^{\mathrm{\infty }}(Y)}}^{H^1(Y)},V_{\mathrm{\#}}:=\{v\in H_{\mathrm{\#}}^1(Y):{\int }_{Y}vdy=0\}\mathrm{.}$ The integral is the normalized cell mean because $\mathrm{\mid }Y\mathrm{\mid }=1$. Derivatives are weak derivatives on $Y$ and the periodic condition is the closure convention in this definition; no pointwise boundary trace is being imposed. Equip $V_{\mathrm{\#}}$ with the inherited $H^1$ norm.

The cell Poincaré inequality on this exact mean-zero space is proved in Mean-zero Poincare inequality on the periodic cell. Its conclusion makes $\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_{L^2(Y)}$ an equivalent complete norm on $V_{\mathrm{\#}}$.