Assume $u\in H^2(\Omega)$ for bounded domain $\Omega\subset\mathbb{R}^2$, in general $\|\nabla u\|_{L^2(\Omega)}\le C\|D^2u\|_{L^2(\Omega)}$ is not true as the trace of the function $\nabla u$ may not be $0$. However, do we have $\|\nabla u\|_{L^2(\Omega)}\lesssim\|D^2u\|_{L^2(\Omega)}+\|u\|_{L^2(\Omega)}$?
One counter example that makes $\|\nabla u\|_{L^2(\Omega)}\le C\|D^2u\|_{L^2(\Omega)}$ fail to be true is $u$ is a general linear function, which means that right hand side is $0$ while the left hand side is not. Now for the second inequality, when right hand side is $0$, the left hand side has to be $0$. Can we then use an usual argument by contradiction used in the proof of Poincare inequality to prove the second inequality?