Regularity assumption for parallel overlapping Maxwell domain decomposition

Let $\Omega\subset\mathbb{R}^3$ be the cylindrical domain

$$\Omega:=\big\{ (x_1,x_2,x_3) \in \mathbb{R}^3 \,|\, x_1 \in [0,L],\,(x_2,x_3)\in \Omega_p \big\},$$

where $\Omega_p\subset\mathbb{R}^2$ is a Lipschitz polygon. Suppose that each partition-of-unity function $\chi_\ell$ depends only on $x_1$ and vanishes in a neighbourhood of $\partial\Omega_\ell\setminus\partial\Omega$. Regularity assumption. Under these conditions, Assumption holds. This provides a setting in which the commutator terms arising in the Maxwell domain decomposition analysis have the regularity required for the convergence proof. The claim is presented as a sufficient geometric and partition-of-unity condition; no resolution status beyond the statement is given.