Let $\Omega$ be a Lipschitz bounded domain of $\mathbb{R}^n$, divided in two Lipschitz subdomains $\Omega_1$ and $\Omega_2$ such that $\Omega_1 \cap \Omega_2 = \emptyset$. We define the following boundaries: $$ \Gamma = \partial \Omega_1 \cap \partial \Omega_2,\\ \Gamma_1 = \partial \Omega_1 \setminus \Gamma. %\Gamma_2 = \partial \Omega_2 \setminus \Gamma. $$

Let $u$ be in $H^n(\Omega_1)$ such that $u_{|\Gamma_1} = 0$.

I wonder if there exists an extension of $u$ in $H^n(\Omega)$, denote by $v$, such that $v_{|\partial \Omega}=0$ ?