Zero facet traces of functions in H1_0 on an affine mesh

Let $F_K(\widehat{x})=b_K+A_K\widehat{x}$ be the reference map and let $\widehat{w}=w\circ F_K$. By [F2], $\widehat{w}\in W^{1,2}({\widehat{S}}_d)$ and its norm is bounded by a finite constant depending on $A_K$ times $\mathrm{\parallel }w{\mathrm{\parallel }}_{W^{1,2}(K)}$. The surface Jacobian $J_F$ is positive and constant by [F3]. Define $T_{K,F}w$ by pullback of ${\gamma }_{\widehat{F}}\widehat{w}$; then $\mathrm{\parallel }{T}_{K,F}w{\mathrm{\parallel }}_{L^2(F)}^2=J_F\mathrm{\parallel }{\gamma }_{\widehat{F}}\widehat{w}{\mathrm{\parallel }}_{L^2(\widehat{F})}^2\le J_F{C}_{\widehat{F}}^2\mathrm{\parallel }\widehat{w}{\mathrm{\parallel }}_{W^{1,2}({\widehat{S}}_d)}^2\le C_{K,F}^2\mathrm{\parallel }w{\mathrm{\parallel }}_{W^{1,2}(K)}^2\mathrm{.}$ Thus the trace is bounded, and it agrees with ordinary restriction for smooth $w$ by [F1].

Let $w\in H_0^1(\mathrm{\Omega };\mathbb{R})$. For each $m\ge 1$, the closure property [F4] and Countable Choice [F5] give ${\varphi }_m\in C_c^{\mathrm{\infty }}(\mathrm{\Omega };\mathbb{R})$ with $\mathrm{\parallel }w-{\varphi }_m{\mathrm{\parallel }}_{W^{1,2}(\mathrm{\Omega })}<1\mathrm{/}m$. Restriction to $K$ cannot increase this norm, so step 1.1 gives $\mathrm{\parallel }{T}_{K,F}(w{\mathrm{\mid }}_K)-T_{K,F}({\varphi }_m{\mathrm{\mid }}_K){\mathrm{\parallel }}_{L^2(F)}\le C_{K,F}\mathrm{/}m\mathrm{.}$ The support of each ${\varphi }_m$ is compactly contained in $\mathrm{\Omega }$, hence ${\varphi }_m$ vanishes in a neighborhood of the boundary facet $F$ and its trace is zero by step 1.1. Letting $m\to \mathrm{\infty }$ proves $T_{K,F}(w{\mathrm{\mid }}_K)=0$.

If $p$ is polynomial on $K$, then it is smooth and step 1.1 identifies $T_{K,F}(p)$ with its classical restriction. Therefore for a global piecewise-polynomial $p\in H_0^1(\mathrm{\Omega })$, step 2.1 makes that restriction zero on each boundary facet. This proves the final assertion.