Let $D$ be a bounded domain of $\mathbb{R}^n$ with smooth boundary $\partial D$. Is the following subspace dense in space $L^2(D)\times L^2(\partial D)$:
$$\{(f,f\rvert\_{\partial D}) : f\in C^\infty(\overline{D}), (\Delta f)\rvert\_{\partial D}=0\}.$$
I tried to use density of $C\_c^\infty(D)$ in $L^2(D)$, but I didn't get a final answer. This question was motivated by the density of the subspace
$$\{(f,f\rvert\_{\partial D}) : f\in C^\infty(\overline{D})\}.$$
Thank you for any hint.



Density of a functional space
I believe that Daniele Tampieri's idea of using completeness Fichera's Theorem is the right one. However I think there is a simpler proof.
In view of the Hahn-Banach Theorem, proving the density of the space
\begin{equation}
\left\{(f,f|\_{\partial D})\ |\ f\in S \right\}, \quad \text{where } 
S=\left\{ f\in C^{\infty}(\overline{D})\ |\ \Delta f|\_{\partial D} =0 \right\},
\end{equation}
in $L^{2}(D)\times L^{2}(\partial D)$ 
is equivalent to show that, if $(F,G)\in L^{2}(D)\times 
L^{2}(\partial D)$ is such that
\begin{equation}
\int\_{D}F\, f\, dx + \int\_{\partial D}G\, f\, d\sigma = 0
\end{equation}
for any $f\in S$, then $F=0$ a.e. in $D$
and $G=0$ a.e. on $\partial D$.
So suppose that these orthogonality conditions hold. In particular we have
$$
\int\_{D}F\, f\, dx = 0
$$
for any $f\in C^{\infty}\_{c}(D)$. It is well known that this implies
$F=0$ a.e. in $D$. 
Therefore
$$
\int\_{\partial D}G\, f\, d\sigma = 0
$$
for any $f\in S$. Harmonic polynomials obviously belong to $S$ and then
$$
\int\_{\partial D}G\, \omega\, d\sigma = 0
$$
for any harmonic polynomial $\omega$. Fichera's theorem implies
 $G=0$ a.e. on $\partial D$.


