Let $\Omega\subset\mathbb{R}^n$ be a bounded open set and $u\in W_0^{1,2}(\Omega)$. Define $B_R=B(x_0,R)$ for $x_0\in\partial\Omega$ and consider $\tilde{u}=u\chi_{\Omega\cap B_{2R}}$. Do we need some extra assumptions in order to prove that $\tilde{u}\in W_0^{1,2}(B_{3R})$?
I am doing this because I want to bound
$$ \Big(\int_{\Omega\cap B_{2R}}u^2\Big)^{1/2}\leq C\Big(\int_{\Omega\cap B_{2R}}|\nabla u|^{2_*}\Big)^{1/2_*} $$ using Sobolev-Poincaré's inequality, but it is not true that $u\in W_0^{1,2}(\Omega\cap B_{2R})$.
Thank you very much for your help.