An inequality for $W^{k,p}$ norms Let $u \in W_0^{2,p}(\Omega)$, for $\Omega$ a bounded subset of $\mathbb R^n$. I am trying to obtain the bound
$$\|Du\|_p \leq \epsilon \|D^2 u\|_p + C_\epsilon \|u\|_p$$
for any $\epsilon > 0$ (here $C_\epsilon$ is a constant that depends on $\epsilon$, and $\|.\|_p$ is the $L^p$ norm). I tried deducing this from the Poincare inequality, but that does not seem to get me anywhere. I also tried proving the one dimensional case first, but was no more able to do that than the $L^p$ case. Any suggestions for how to proceed with this problem?

Such inequalities appear all over the place in PDE theory. They all can be seen as instances of Ehrling's lemma. Here, you have
$$ (W^{2,p}_0(\Omega), ||\;||_3) \hookrightarrow (W^{1,p}_0(\Omega), ||\;||_2) \hookrightarrow (L^p(\Omega), ||\;||_1) $$
where
$$ ||u||_3 = ||D^2u||_p, ||u||_2 = ||Du||_p, ||u||_1 = ||u||_p. $$
The first inclusion is compact, the second continuous and hence from Ehrling's lemma you have for any $\epsilon > 0$ a constant $C(\epsilon) > 0$ such that
$$ ||u||_2 \leq \epsilon ||u||_3 + C(\epsilon)||u||_1. $$
The fact that $||\;||_2$ is an equivalent norm for the Sobolev space $W^{1,p}_0(\Omega)$ is the Poincaré inequality. The fact that $||\;||_3$ is an equivalent norm for the Sobolev space $W^{2,p}_0(\Omega)$ can itself be seen as an application of Ehrling's lemma together with the Poincaré inequality.
