Poincaré-like inequality

Question

Let $\Omega\subset\mathbb{R}^3$ be an open bounded set. Let $\partial\Omega=\Gamma^1\cup\Gamma^2$, with $\Gamma^1\cap\Gamma^2=\emptyset$. We denote as $\Gamma^1_j$, $j=1,\dots,p_{\Gamma^1}+1$, the connected components of $\Gamma^1$. Define $$ H^1_\ast(\Omega):=\{ \xi\in H^1(\Omega):\xi_{|{\Gamma^1_j}}\,\mbox{is constant}\,\forall\ j=1,\dots,p_{\Gamma^1}, \xi_{|\Gamma^1_{p_{\Gamma^1}+1}}=0\}. $$ I read in some articles that Poincaré inequality holds for functions in $H^1_\ast(\Omega)$, i.e. there exists $C>0$ such that $$ ||\xi||_{L^2(\Omega)}\le C ||\nabla\xi ||_{L^2(\Omega)}, $$ for every $\xi\in H^1_\ast(\Omega)$. Any hint?