Density in higher order Sobolev space

Question

Let $ 1 < p < + \infty $ and $ \Omega $ be a smooth domain of $ \mathbb{R}^N,\ N \geq 2. $ Is it true that $ C_c^{\infty}(\Omega) $ is dense into the "intermediate space" $ W^{2,p}( \Omega) \cap W_0^{1,p}( \Omega) ? $

Thanks in advance.

Answer 1

No, the closure of $C_c^\infty(\Omega)$ w.r.t.\ the norm of $W^{2,p}(\Omega)$ is (by definition) $W_0^{2,p}(\Omega)$ and this space is strictly included in $W^{2,p}(\Omega) \cap W^{1,p}_0(\Omega)$.