Functional analysis – Density of subspaces of \$ L ^ 2 \$

Leave $$D$$ be a bounded domain of $$mathbb {R} ^ n$$ with smooth limit $$partial D$$. How can you verify 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 $$D_1$$ It is dense while $$D_2$$ It is not. I tried to use the density of the test functions but I don't get the final result.

Thanks for any suggestions.