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Let $\Omega$ be a sufficiently regular domain, for example $\Omega=B(0,r)$, $r>0$, and $m(x)=\textrm{dist}(x,\partial \Omega)$ be the distance to boundary function.

Suppose $\mathcal{F}$ is a given family of smooth functions, which is dense in $C^m(\bar\Omega)$ (with corresponding $m$-th order sup-norms) for arbitrary $m$. Is it possible to explicitly construct another family $\mathcal{G}$ of functions based on $\mathcal{F}$ and the function $m(x)=\textrm{dist}(x,\partial \Omega)$, such that $\mathcal{G}$ is dense in $W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$ (i.e., any function $g\in \mathcal{G}$ satisfies that $g|_{\partial \Omega}=0$)?

A natural candidate is the set $$ \mathcal{G}=\{ mf\mid f\in \mathcal{F}\}, $$ where I directly multiply the function $f$ by the distance function. Since $\Omega$ has a sufficient regular boundary, the function $m$ is sufficiently smooth, hence for any $f\in C^{2}(\bar{\Omega})$, $mf\in C^2(\bar{\Omega})\cap C_0(\bar{\Omega})$. I also know $C^{2}(\bar{\Omega})\cap C_0(\bar{\Omega})$ is dense in $W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$ (in $W^{2,2}$ norm). But I was not able to proceed further to show $\mathcal{G}$ is dense in $C^2(\bar{\Omega})\cap C_0(\bar{\Omega})$.

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This is a partial answer (too long for a comment). The idea is to reduce to the case when the normal derivative of $f$ vanishes on the boundary.

In the following let us use the notation from the book Sobolev spaces by R.A. Adams. In particular, $\|f\|_{s,p,\Omega}$ is the norm of $f$ in $W^{s,p}(\Omega)$, let $T= \prod_{k=0}^{s-1} W^{s-k-1/p, p}(\partial \Omega)$ and let $\gamma \colon W^{s,p}(\Omega) \to T$ denote the trace map, i.e. the extension from $C_0^\infty(\mathbb{R}^n)$ to $W^{s,p}(\Omega)$ of the map $f \mapsto (\gamma_0(f),\dots, \gamma_{s-1}(f)) := (f|_{\partial \Omega}, \dots, \frac{\partial^{s-1} f}{\partial n^{s-1}}|_{\partial \Omega})$, for more details see Section 7.52 in the cited book. In our case $s=p=2$. By theorem 7.53 the operator $\gamma$ is bounded and for any $g\in T$ there exists $E(g)\in W^{s,p}(\Omega)$ such that $\gamma(E(g)) = g$ and $\|E(g)\|_{s,p,\Omega} \le C \|g\|_T$. (All constants are denoted below with $C$.)

Let $f\in W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$. Take $\phi \in \mathcal{F}$ such that $\|f - \phi\|_{2,2,\Omega} < \epsilon$.

Define $\tilde \phi:=E(0,\gamma_1(\phi))$, i.e. take a function $\tilde\phi$ with the same normal derivative as $\phi$ but zero value on $\partial \Omega$.

Then $\gamma_0(f-\tilde \phi)=0$ and $\|\gamma_1(f-\tilde \phi)\| = \|\gamma_1(f - \phi)\| \le C \epsilon$. Hence there exists a function $r = E(\gamma(f-\tilde \phi))\in W^{2,2}(\Omega)$ such that $\gamma(r) = \gamma(f-\tilde \phi)$ and $\|r\| \le C \epsilon$. Therefore $\gamma(f - \tilde \phi - r) = (0,0)$, that is $f-\tilde \phi-r \in W^{2,2}_0(\Omega)$. Such functions can easily be approximated by functions of the form $\zeta \cdot \psi$, where $\zeta$ is a cutoff function (0 at $\partial \Omega$, 1 sufficiently far from $\partial \Omega$) and $\psi \in \mathcal{F}$.

Thus we have written $f$ in the form $f = \tilde \phi + \zeta \cdot \psi + \tilde r$, with $\tilde \phi + \zeta \cdot \psi \in W^{2,2}(\Omega)\cap W^{1,2}_0(\Omega)$ and $\|\tilde r\|_{2,2,\Omega}$ is small. Therefore $$ \mathcal{G} = \{\tilde \phi + \zeta\cdot \psi: \phi,\psi\in \mathcal{F}, \text{ $\zeta$ boundary cutoff function}\} $$ where $\tilde \phi = E(0,\gamma_1(\phi))$.

If one were able to write $E(0,\gamma_1(\phi))$ explicitly using the distance to the boundary (I believe this is possible), then this construction would be more explicit.

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