Extension of $H_0^1$ functions

Question

Let $\Omega_0\subseteq \Omega_1$, with $\Omega_0,\Omega_1$ bounded and open subsets of $\mathbb{R}^N$. Let us suppose that $f\in L^2(\mathbb{R}^N)$ and $\text{supp} f \subseteq \Omega_0$. Consider the unique solutions $u_0\in H_0^1(\Omega_0),u_1\in H_0^1(\Omega_1)$ of the following variational problems $$\int_{\Omega_0} \nabla u_0\cdot \nabla \varphi=\int_{\Omega_0} f \varphi \qquad \forall\varphi \in H_0^1(\Omega_0)$$ and $$\int_{\Omega_1} \nabla u_1\cdot \nabla \varphi=\int_{\Omega_1} f \varphi \qquad \forall\varphi \in H_0^1(\Omega_1)$$ I was wondering if the following statement is true: The function $u_1$ is the extension by zero of $u_0$ to $\Omega_1$.

Answer 1

Note that

$$ \int_{\Omega_1} \nabla (\tilde{u}_0−u_1)\cdot \nabla \varphi=0, $$ where $\tilde{u}_0$ is the extension of $u_0$ by $0$ on $\Omega_1\setminus \Omega_0$. As this holds for all $\varphi \in H^1_0(\Omega_1)$, it follows that $\tilde{u}_0$ and $u_1$ differ only by a constant, which must be $0$ by the boundary conditions.