Mean-zero Poincare inequality on the periodic cell

For smooth periodic $v$ and each coordinate $0\le i<d$, opposite-face values agree, so integrating that coordinate derivative over $Y$ and applying the one-dimensional FTC [F2] gives ${\int }_Y{\mathrm{\partial }}_{i}v=0$. Approximate an arbitrary $v\in H_{\mathrm{\#}}^1(Y)$ in $H^1$ by smooth periodic functions using [F1]; convergence of the derivatives in $L^2(Y)$ implies convergence of their integrals by [F3] and $\mathrm{\mid }Y\mathrm{\mid }=1$. Thus every component of ${\int }_Y\mathrm{\nabla }v$ is zero.

For smooth periodic $v$, write $m={\int }_{Y}v(y)dy$. For $x,y\in Y$, telescope from $y$ to $x$ one coordinate at a time, in order $0,\dots ,d-1$. The $i$th difference is the integral of ${\mathrm{\partial }}_{i}v$ along a segment of length at most one, with the earlier coordinates fixed at $x$ and later coordinates fixed at $y$. Cauchy--Schwarz [F3] bounds its square by the integral of $\mathrm{\mid }{\mathrm{\partial }}_{i}v{\mathrm{\mid }}^2$ along the full unit coordinate interval. The inequality $\mathrm{\mid }{\sum }_{i<d}{b}_i{\mathrm{\mid }}^2\le d{\sum }_{i<d}\mathrm{\mid }{b}_i{\mathrm{\mid }}^2$ and Tonelli [F4], integrating in $x$ and $y$, then give ${\int }_Y{\int }_Y\mathrm{\mid }v(x)-v(y){\mathrm{\mid }}^{2}dydx\le d{\sum }_{i<d}\mathrm{\parallel }{\mathrm{\partial }}_{i}v{\mathrm{\parallel }}_2^2\mathrm{.}$ For each $x$, Jensen's inequality gives $\mathrm{\mid }v(x)-m{\mathrm{\mid }}^2=\mathrm{\mid }{\int }_Y(v(x)-v(y))dy{\mathrm{\mid }}^2\le {\int }_Y\mathrm{\mid }v(x)-v(y){\mathrm{\mid }}^{2}dy$. Integrating in $x$ proves the stated inequality for smooth periodic mean-zero $v$.

Approximate any $v\in V_{\mathrm{\#}}$ in $H^1$ by smooth periodic $v_n$ using Countable Choice [F5]. Their means converge to zero by Cauchy--Schwarz, so $v_n-{\int }_Y{v}_n$ are smooth periodic mean-zero functions converging to $v$ in $H^1$. Passing to the limit in step 1.2 proves the inequality on $V_{\mathrm{\#}}$. By [F7], $H^1(Y)$ is complete. Its closure-defined subspace $H_{\mathrm{\#}}^1(Y)$ is closed, and the mean is continuous by $\mathrm{\mid }{\int }_{Y}v\mathrm{\mid }\le \mathrm{\parallel }v{\mathrm{\parallel }}_2\le \mathrm{\parallel }v{\mathrm{\parallel }}_{H^1}$, so $V_{\mathrm{\#}}$ is closed in $H^1(Y)$. Every $H^1$-Cauchy sequence in $V_{\mathrm{\#}}$ therefore has an ambient limit in $V_{\mathrm{\#}}$, proving completeness of its inherited norm. The inequality shows the gradient norm and $H^1$ norm are equivalent, hence the gradient norm is complete too.