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

  • $\begingroup$ $U$ is bounded? The classical Poincaré inequality only need $u\in W_0^{1,p}(U)$. $\endgroup$ – user10354138 May 13 at 1:22
  • $\begingroup$ I am considering even the case when $U$ is unbounded. $\endgroup$ – Casey May 13 at 1:25
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    $\begingroup$ 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 $\varphi_n(x)=\varphi(c_n+x/n)$. $\endgroup$ – user10354138 May 13 at 1:59
  • $\begingroup$ Sorry, but what are $\varphi$ and $c_n$? $\endgroup$ – Casey May 13 at 2:15
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    $\begingroup$ 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 $U$. $\endgroup$ – user10354138 May 13 at 2:16

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