I have a bounded Lipschitz domains $\Omega, \Omega_1, \Omega_2 \subset \mathbb{R}^3$ such that $\overline{\Omega}=\overline{\Omega}_1 \cup \overline{\Omega}_2$ and $\Omega_1 \cap \Omega_2=\emptyset$. Let $u \in H^k(\Omega)$ and $v \in C^\infty(\Omega)$. When is the following true:
$ (u,v)_{L^2(\Omega)} = (u,v)_{L^2(\Omega_1)} + (u,v)_{L^2(\Omega_2)} $
From Sobolev embedding theorem, for $k \geq 2$, $u$ is continuous and the above is true.
For $k=-1$, I can construct counter examples (for example, gradient of a discontinuous function).
What about for $k=0$ and $k=1$?
This is always true, as $\Omega \setminus (\Omega_1\cup \Omega_2)$ can only contain the boundaries of $\Omega_1$ and $\Omega_2$, which have zero measure due to the regularity assumptions on the domains.