Let $u\in H^1_0(\Omega)$, where $\Omega$ is a bounded open set in $\Bbb{R}^2$. Is the following true:
For some $\alpha>0$, $\int_{\Omega} u^2\leq \alpha \int_{\Omega} |Du|^2$
I remember reading this somewhere, but can't quite recall where. Note that $u\equiv 0$ at the boundary. I think I read it in Evans' book but can't find it now