I am trying to understand the Corollary 9.19 (Poincaré’s inequality) in Functional Analysis, Sobolev Spaces and Partial Differential Equations, by Haim Brezis.
Suppose that $1 \le p < \infty$ and $\Omega$ is a
bounded open set. Then there exists a constant $C$ (depending on $\Omega$ and $p$) such that $\left\Vert u \right\Vert_{L^p(\Omega)} \le C \left\Vert \nabla u \right\Vert_{L^p(\Omega)}$ for all $u \in W^{1,p}_0(\Omega)$.
So far, I know that the zero extension $\bar u$ of $u\in W^{1,p}_0(\Omega)$ is an element in $W^{1,p}(\mathbb{R}^N)$ (which is a consequence of the preceding Proposition 9.18).
Case 1: $1\le p<N$
With the help of Theorem 9.9 (Sobolev, Gagliardo, Nirenberg),
$\left\Vert \bar u \right\Vert_{L^{p^*}(\mathbb{R}^N)} \le C \left\Vert \nabla \bar u \right\Vert_{L^p(\mathbb{R}^N)}$, where $\frac{1}{p*} = \frac{1}{p} – \frac{1}{N}$,
I can show the Poincaré’s inequality due to compact support and $p<p^*$.
Case 2: $p>N$
Theorem 9.12 (Morrey) asserts $|\bar u(x) – \bar u(y)|\le C |x-y|^\alpha \left\Vert \nabla \bar u \right\Vert_{L^p(\mathbb{R}^N)} $ a.e.
Take the continuous representative and still denote $\bar u$. By translation, I may take $y = 0$ and $\bar u(y) = 0$. Then taking p-norm (or $\infty$-norm) yields the result.
My question is how to conclude the Poincaré’s inequality when $p=N$?
Best Answer
For any $1\le p_1<N$ we have the embedding $$ W^{1,N}_0 (\Omega) \to W^{1,p_1}_0 (\Omega)$$ since $\Omega$ is bounded. Now choose $p_1 <N$ so that
$$ p_1^* = \frac{p_1N}{N-p_1}> N. $$
Then there is a compact embedding
$$ W^{1,N}_0 (\Omega) \to W^{1,p_1} (\Omega) \to L^N (\Omega).$$
(the second $\to$ is compact). Then for all $u\in W^{1,N}_0(\Omega)$,
$$ \| u\|_{L^N} \le C \|u\|_{L^{p_1^*}} \le C \| \nabla u\|_{L^{p_1}} \le C \| \nabla u\|_{L^N}.$$