A bounded regular decomposition with both summands satisfying full Dirichlet conditions

Fix a ball $B$ with $\bar{\mathrm{\Omega }}⋐B$, let $U$ be the zero extension of $u$ by [F1], and use [F2] with support $\bar{\mathrm{\Omega }}$. Then $U=w+\mathrm{\nabla }p$, $\mathrm{\parallel }w{\mathrm{\parallel }}_{H^1}\le C\mathrm{\parallel }u{\mathrm{\parallel }}_{H(\mathrm{curl})}$, and $\mathrm{\parallel }p{\mathrm{\parallel }}_{H^1}\le C_{\mathrm{\Omega }}\mathrm{\parallel }u{\mathrm{\parallel }}_2$. The exterior open set $D=B\setminus \bar{\mathrm{\Omega }}$ has Lipschitz boundary: near $\mathrm{\partial }\mathrm{\Omega }$ use the same graph with the opposite side, and near $\mathrm{\partial }B$ use the smooth sphere; these boundary sets are a positive distance apart. It may have several components, which [F3] permits. On $D$, $U=0$, so $\mathrm{\nabla }p=-w$. Hence $p{\mathrm{\mid }}_D\in H^2(D)$, with $\mathrm{\parallel }p{\mathrm{\parallel }}_{H^2(D)}\le C_{\mathrm{\Omega }}\mathrm{\parallel }u{\mathrm{\parallel }}_{H(\mathrm{curl})}$. This follows directly from the definition of second weak derivatives: $D_i{D}_{j}p=-D_i{w}_j\in L^2(D)$.

Extend $p{\mathrm{\mid }}_D$ by [F3] to $a=E_D(p{\mathrm{\mid }}_D)\in H^2({\mathbb{R}}^3)$. Choose one fixed $\chi \in C_c^{\mathrm{\infty }}(B)$ equal to one near $\bar{\mathrm{\Omega }}$, and define whole-space fields $Z=\chi (w+\mathrm{\nabla }a)$ and $P=\chi (p-a)$. Both are in $H^1({\mathbb{R}}^3)$ with norms bounded by $C_{\mathrm{\Omega }}\mathrm{\parallel }u{\mathrm{\parallel }}_{H(\mathrm{curl})}$. On $D$, $a=p$ and $\mathrm{\nabla }a=\mathrm{\nabla }p=-w$ almost everywhere, so $Z=P=0$ there; outside $B$ their cutoff makes them zero. Thus they vanish almost everywhere outside $\bar{\mathrm{\Omega }}$. By [F1], $z=Z{\mathrm{\mid }}_{\mathrm{\Omega }}\in H_0^1(\mathrm{\Omega };{\mathbb{R}}^3)$ and $q=P{\mathrm{\mid }}_{\mathrm{\Omega }}\in H_0^1(\mathrm{\Omega })$. On $\mathrm{\Omega }$, $\chi =1$ and its gradient is zero, giving $z+\mathrm{\nabla }q=w+\mathrm{\nabla }a+\mathrm{\nabla }p-\mathrm{\nabla }a=u$. Fixed zero extension, Fourier multipliers, $E_D$, and cutoff make the maps $R:u\mapsto z$, $Q:u\mapsto q$ linear; their estimates have just been proved. Only the geometry of $\mathrm{\Omega },B$, not a mesh or a topological assumption, enters the constants.