How can I prove this inequality
$$ \parallel \nabla u \parallel_{H^1(\Omega)}^2 \leq C (\parallel \nabla u \parallel_{L^2(\Omega)}^2+\parallel \Delta u \parallel_{L^2(\Omega)}^2), $$
where $u\in H^2(\Omega)$ such that $\frac{\partial u }{\partial \nu}=0$ on $ \partial \Omega$.
The $\nabla (\nabla u)$ is it $D^2 u$?.
This is Theorem 3.4 together with a density argument in The weak Dirichlet and Neumann problem for the Laplacian in $L^p$ for bounded and exterior domains. Applications. in Nonlinear Analysis, Function Spaces and Applications, volume 4, Vieweg+Teubner Verlag, Wiesbaden, 1990, pp.~180--223 by Christian Simander
This can be found for example via your favorite online library.