A Laplacian inequality over a closed domain in R^3

Question

Consider a 3D-domain $\Omega = \left\{(x,y,z): 0 \leq x \leq a, 0 \leq y \leq b, 0 \leq z \leq 1\right\}$. Let $u(x) \in H^2(\Omega)$ be a function which is zero at the boundary $\partial\Omega$. The inequality for the supremum of a function is well-known: $$\sup\limits_\Omega \vert{u(x)}\vert \leq C_* \lvert\lvert{\nabla^2 u}\lvert\lvert.$$ It is asserted that there exists $\alpha \geq \lambda_1 > 0$ such that $$\lvert\lvert{\nabla^2 u}\lvert\lvert \leq \alpha \vert \vert u \vert \vert.$$ Where $\lambda_1$ is the first eigenvalue of Dirichlet Laplacian. Above two inequalities will then imply that $$\sup\limits_\Omega \vert{u(x)}\vert \leq \alpha C_* \lvert\lvert{u}\lvert\lvert. $$ Is it correct ?