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?
, we can try an argument by contradiction, and using reflexivity of $L^p(\Omega)$.
– 2012-11-10