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.