Density of subspaces of $L^2$

Question

Let $D$ be a bounded domain of $\mathbb{R}^n$ with smooth boundary $\partial D$. How can check if the following spaces are dense (or not) in $L^2(D)$.

$$D_1:=\{f\in H^3(D): f\rvert_{\partial D}=(\Delta f)\rvert_{\partial D}=0\}.$$ $$D_2:=\{f\in H^3(D): (\Delta f)\rvert_{\partial D}=0\}.$$ I think $D_1$ is dense while $D_2$ is not. I tried to use density of test functions but I don't get the final result.

Thank you for any suggestion.

Answer 1

The set of smooth functions with compact support $C_c^\infty(D)$ is a subset of both $D_1$ and $D_2$ and it is dense in $L^2(D)$. Hence, $D_1$ and $D_2$ are dense in $L^2(D)$ as well.