Does the Poincaré inequality $$\int_{\Omega} |u|^p dx\leq C \int_{\Omega} |\nabla u|^p dx$$ hold if we do not have $u\in W^{1,p}_0$, but have $u\in W^{1,p}$ and $u(x_0)=0$ in trace sense at some $x_0\in\partial\Omega$ or $u\in W^{1,p}$ and $u(x)=0$ in a small neighborhood of some $x_0\in\partial\Omega$?
$\Omega$ is an open bounded set in $\Bbb R^n$.
You may adapt the proof of Wirtinger's inequality for real functions only by assuming $u(x)=0$ almost everywhere on $\partial \Omega$ (and you probably need some extra assumption on the regularity of $\partial\Omega$). $u(x_0)=0$ is an empty condition, because functions in $\mathcal{L}^p(\Omega)$ do not need to be pointwise-defined.