# Poincaré's inequality proof for $u \in W_0^{1,p}(\Omega)$.

I am trying to prove Poincare's inequality for $u \in W_0^{1,p}(\Omega)$, where $\Omega \subset \mathbb{R}^n$ is an open bounded set and $1 \leq p < \infty$. This is Poincare's inequality:

$||u||_{L^p(\Omega)} \leq C ||\nabla u ||_{L^p(\Omega)}$.

For $p < N$ the inequality follows applying Sobolev-Gagliardo-Nirenberg inequality. Indeed, $$||u||_{p^*} \leq C||\nabla u||_{p},$$ and from the fact that $\Omega$ is bounded we have $||u||_p \leq \tilde{C} ||u||_{p^*} \leq C||\nabla u||_{p}$.

How can I prove the inequality for $N \leq p < \infty$ ?

• It is proven in Evans, using a contradiction argument and Rellich's theorem. – Ian Jun 24 '16 at 23:17
• @Ian Can it be proven without using Rellich's theorem? – D1X Jun 24 '16 at 23:19
• I doubt it; the actual value of the constant is quite sensitive to specifics of the domain. – Ian Jun 24 '16 at 23:30
• Okay, thank you very much. – D1X Jun 27 '16 at 12:28