Smooth functions are dense in finite-order Sobolev spaces on a reference simplex

Let $a$ be the barycenter of ${\widehat{S}}_d$ and, for $0<t<1$, set $F_t(x)=a+t(x-a)$. The set $S_t:=F_t^{-1}({\widehat{S}}_d)$ is open and contains $\overline{{\widehat{S}}_d}$: each point of the closure is carried to a strict convex combination of that point and the interior point $a$, which lies in the open simplex. Thus $u_t:=u\circ F_t$ is defined on $S_t$ for $u\in W^{k,p}({\widehat{S}}_d)$, and [F1] gives $D^{\alpha }{u}_t=t^{\mathrm{\mid }\alpha \mathrm{\mid }}(D^{\alpha }u)\circ F_t$.

For every $g\in L^p({\widehat{S}}_d)$, $g\circ F_t\to g$ in $L^p({\widehat{S}}_d)$ as $t\uparrow 1$. To see this, extend $g$ by zero to ${\mathbb{R}}^d$ and approximate that extension in $L^p$ by $\varphi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^d)$ using [F2]. For $t\ge 1\mathrm{/}2$, change of variables bounds the pullback error by $t^{-d\mathrm{/}p}\mathrm{\parallel }g-\varphi {\mathrm{\parallel }}_{L^p({\mathbb{R}}^d)}\le 2^{d\mathrm{/}p}\mathrm{\parallel }g-\varphi {\mathrm{\parallel }}_p$; for fixed $\varphi$, uniform continuity on the compact closure gives $\varphi \circ F_t\to \varphi$ uniformly there. First choosing $\varphi$ with arbitrarily small error and then taking $t\uparrow 1$ proves the claim. Applying it to each of the finitely many $D^{\alpha }u$ and using [F1] shows $u_t{\mathrm{\mid }}_{{\widehat{S}}_d}\to u$ in $W^{k,p}({\widehat{S}}_d)$.

Fix $t<1$. Since $\overline{{\widehat{S}}_d}$ is compactly contained in $S_t$, choose $\eta \in C_c^{\mathrm{\infty }}(S_t)$ equal to one near that closure by [F3]. Then $\eta u_t\in W^{k,p}(S_t)$ has compact support in $S_t$, so its zero extension $U_t$ belongs to $W^{k,p}({\mathbb{R}}^d)$ by [F3]. By [F2], choose ${\varphi }_j\in C_c^{\mathrm{\infty }}({\mathbb{R}}^d)$ converging to $U_t$ in that norm. Since $U_t=u_t$ on a neighbourhood of the closed simplex, the restrictions ${\varphi }_j{\mathrm{\mid }}_{{\widehat{S}}_d}$ converge to $u_t{\mathrm{\mid }}_{{\widehat{S}}_d}$ in $W^{k,p}({\widehat{S}}_d)$.

Given $\epsilon >0$, choose $t$ close enough to one that $\mathrm{\parallel }{u}_t-u{\mathrm{\parallel }}_{W^{k,p}({\widehat{S}}_d)}<\epsilon \mathrm{/}2$ by step 2.1, and then choose $j$ in step 2.2 with $\mathrm{\parallel }{\varphi }_j-u_t{\mathrm{\parallel }}_{W^{k,p}({\widehat{S}}_d)}<\epsilon \mathrm{/}2$. The triangle inequality gives $\mathrm{\parallel }{\varphi }_j-u{\mathrm{\parallel }}_{W^{k,p}({\widehat{S}}_d)}<\epsilon$. The choices are finite for each $u,\epsilon$; Countable Choice is only the declared background of [F4], and no $p=\mathrm{\infty }$ density is used.