Discrete Sobolev Poincare inequality proof in Evans book

Question

In page 275 of Evans book, the Poincare's inequality has been proven via contradiction. I was wondering how can one extend this prove to prove Sobolev-Poincare inequality: $||u-u_\Omega||_{L^{p*}}\le C||Du||_{L^p(\Omega)}$ where $u\in W^{1,p}(\Omega)$, $1\le p<n$, $u_\Omega ={\frac 1\Omega}\int_\Omega u$ and $p^*={\frac {np}{n-p}}$ where $C$ is independent of $u$?

Answer 1

For this question, you only need to use Sobolev embedding.

Once we have $\|u-u_\Omega\|_{L^p}\leq C\|Du\|_{L^p}$, we have $u-u_\Omega\in W^{1,p}(\Omega)$. Hence, the sobolev embedding tells you that $$ \|u-u_\Omega\|_{L^{p^*}} \leq C\| D(u-u_\Omega)\|_{L^p}=C\|Du\|_{L^p} $$ Done.