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I am attempting to demonstrate that for $u \in W^{2,p}(U) \cap W^{1,p}_0(U)$ \begin{align*} \int_U |Du|^p \ dx \leq C\int_U |D^2u|^p \ dx \end{align*}

where $U$ is some open subset of $\mathbb{R}^n$.

All that I have been able to deduce is that \begin{align*} \int_U |Du|^p \ dx \leq C \int_U |u||Du|^{p - 2} |D^2 u| \ dx \end{align*}

by using integration by parts, and I have been unsuccessful with the general Hölder's inequality to demonstrate the required inequality.

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