I am attempting to demonstrate that for all $u \in W^{2,p}(U) \cap W^{1,p}_0(U)$ \begin{align*} \int_U |u|^p \ dx \leq C\int_U |Du|^p \ dx \end{align*}
where $U$ is some open subset in $\mathbb{R}^n$.
It is easy to demonstrate that $|| u ||_{L^{p^*}(U)} \leq C ||Du||_{L^p(U)}$ using the Gagliardo-Nirenberg-Sobolev inequality since $u \in W^{1,p}_0(U)$, but it isn't apparent to me how $u \in W^{2,p}(U) \cap W^{1,p}_0(U)$ implies $|| u ||_{L^{p}(U)} \leq C ||Du||_{L^p(U)}$.