Norm Equivalence in Sobolev Space

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 $$|| u ||_{L^{p}(U)} \leq C ||Du||_{L^p(U)}$$.

• $U$ is bounded? The classical Poincaré inequality only need $u\in W_0^{1,p}(U)$. – user10354138 May 13 at 1:22
• I am considering even the case when $U$ is unbounded. – Casey May 13 at 1:25
• Then you need some constraints on $U$. For example, if you can fit balls of radius $n$ inside $U$ for every $n$ then it is easy to defeat this by $\varphi_n(x)=\varphi(c_n+x/n)$. – user10354138 May 13 at 1:59
• Sorry, but what are $\varphi$ and $c_n$? – Casey May 13 at 2:15
• Oops, the $c_n$ goes the other way round, $\varphi_n(c_n+x)=\varphi(x/n)$. $\varphi$ is a bump function with support inside the unit open ball, $c_n$ is the center of ball of radius $n$ that you can fit inside $U$. – user10354138 May 13 at 2:16