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?

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Here we dealt with something similar, maybe it can be of some help to you: matematicamente.it/forum/… – Giuseppe Negro Nov 9 '12 at 23:01
Here is the $L^\infty$ version of this result: math.stackexchange.com/questions/207476/… – Lukas Geyer Nov 9 '12 at 23:18
At least, when $1<p<\infty$, we can try an argument by contradiction, and using reflexivity of $L^p(\Omega)$. – Davide Giraudo Nov 10 '12 at 11:22

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.