Let $\Omega \subset \mathbb{R}^2$ be a sufficiently nice domain. It is known that the norm on the space $H^2(\Omega)$ is given by $$||f||^2_2 = ||f||^2_{L^2} + ||\nabla f||^2_{L^2} + ||\Delta f||^2_{L^2}$$ I also know that there is an equivalent norm on $H^2(\Omega) \cap H^1_0(\Omega)$ given by $$||f||^2_2 =||\Delta f||^2_{L^2}$$
My question is what about the norm $$|f|^2_2 = ||\nabla f||^2_{L^2} + ||\Delta f||^2_{L^2}$$
Attempt: Yes, it is equivalent. It is obvious that $$||f||^2_2 = ||f||^2_{L^2} + ||\nabla f||^2_{L^2} + ||\Delta f||^2_{L^2} \geq ||\nabla f||^2_{L^2} + ||\Delta f||^2_{L^2} = |f|^2_2. $$ On the other hand, since we are in $H^1_0(\Omega)$ we can use Poincare inequality with constant $C_p$ and write $$||f||^2_2 = ||f||^2_{L^2} + ||\nabla f||^2_{L^2} + ||\Delta f||^2_{L^2} \leq (1+ C^2_p) ||\nabla f||^2_{L^2} + ||\Delta f||^2_{L^2}$$ Therefore, for some contact $C$ we can have $$||f||^2_2 \leq C |f|^2_2 .$$
From both inequalities we deduce that $||f||^2_2$ and $|f|^2_2$ are equivalent on $H^2(\Omega) \cap H^1_0(\Omega)$.
Is this correct?