Partial extension across an artificial sphere and annular cutoff gluing

Compactness of the artificial sphere gives $\delta >0$ with $v=0$ for $L-\delta <\mathrm{\mid }x\mathrm{\mid }<L$ inside $G_0$; shrink $\delta$ if necessary to separate this collar from $\mathrm{\Theta }$. Choose $\rho \in C_c^{\mathrm{\infty }}(B_L)$ equal to $1$ on $B_{L-\delta }$ and supported in $B_{L-\delta \mathrm{/}2}$. If $E_{G_0}$ is the ordinary simultaneous extension, then $r_{\mathrm{\Omega }}(\rho E_{G_0}v)$ equals $v$ on $G_0$ and zero outside $B_L$, so it is exactly the partial extension. Its $H^s(\mathrm{\Omega })$ norm is at most $C_{s,\rho }\mathrm{\parallel }v{\mathrm{\parallel }}_{H^s(G_0)}$ by the whole-space extension and multiplier bounds. The construction uses values inside $\mathrm{\Theta }$ only as an ordinary extension auxiliary and then restricts to $\mathrm{\Omega }$; it never asserts zero extension across the obstacle. The dependence on $\rho$ is retained, so no uniform bound for arbitrary shrinking gaps is inferred.

For a fixed admissible truncation $G_0$, restriction $H^s(\mathrm{\Omega })\to H^s(G_0)$ is contractive: every whole-space extension of $U$ is also one of $U{\mathrm{\mid }}_{G_0}$, so take the infimum of their norms. F4 on each obstacle-boundary component therefore makes ${\mathrm{\Gamma }}_{\mathrm{\Omega }}$ bounded and linear into $H^{s-1\mathrm{/}2}(\mathrm{\partial }\mathrm{\Theta })$. To compare two admissible truncations $G_a=\mathrm{\Omega }\cap B_{L_a}$ and $G_b=\mathrm{\Omega }\cap B_{L_b}$, take one whole-space $F\in H^s({\mathbb{R}}^n)$ with $F{\mathrm{\mid }}_{\mathrm{\Omega }}=U$, which exists by the restriction-space convention. It extends both $U{\mathrm{\mid }}_{G_a}$ and $U{\mathrm{\mid }}_{G_b}$. The ambient construction in F4 computes both bounded-domain obstacle traces as $T_{\mathrm{\partial }\mathrm{\Theta }}F$, componentwise, so their values agree. This also proves independence of any bounded smooth truncation agreeing with $\mathrm{\Omega }$ on a neighborhood of its compact obstacle boundary. Locality and cutoff commutation follow by restricting to a fixed such truncation and applying those properties in F4; F3 makes the cutoff products bounded. The $L^2$ value agrees with the bounded-domain $H^1$ trace there.

For the gluing formula, $(1-\zeta )u$ has the fixed outer gap $\epsilon$ and bounded $H^s(G_0)$ norm, so the first term is bounded by the preceding construction. Restriction $H^s(G_0)\to H^s(\mathcal{A})$ is contractive in the quotient model, and multiplication by the fixed cutoffs is bounded by extension and whole-space multiplication. The second term is therefore bounded $H^s(G_0)\to H^s(\mathrm{\Omega })$ through $E_{\mathcal{A}}$. These are fixed linear operators. On $\mathcal{A}$ its correction restricts to $\tilde{\zeta }\zeta u=\zeta u$, while below the annulus $\zeta =0$. Hence the two terms sum to $u$ on all of $G_0$.

By step 2.1, $(\mathcal{G}u){\mathrm{\mid }}_{G_0}=u$, so the definition in step 1.2 gives ${\mathrm{\Gamma }}_{\mathrm{\Omega }}\mathcal{G}u={\gamma }_{\mathrm{\partial }\mathrm{\Theta }}^{G_0}u$. Equivalently, near $\mathrm{\partial }\mathrm{\Theta }$ the first term of the gluing formula equals $u$ since $\zeta =0$, and the second term vanishes since $\tilde{\zeta }=0$. The local exterior trace just constructed thus preserves the inner trace, with its higher regularity and its bounded-domain $H^1$ value in $L^2$. No zero extension across $\mathrm{\partial }\mathrm{\Theta }$ is involved.

If the inner trace is zero, choose a smooth $\theta$ equal to $1$ near $\mathrm{\Theta }$ with support inside $B_{L-3\epsilon }$ and put $U=\mathcal{G}u$. The function $w=(\theta U){\mathrm{\mid }}_{G_0}=\theta u$ belongs to $H^s(G_0)$ and has zero trace on both boundary components of $G_0$: on the inner one by hypothesis and on the outer one because $\theta$ vanishes nearby. To place $w$ in the weak space, take a whole-space $H^s$ extension $Q$ of $w$. The Fourier weights imply $Q\in H^1({\mathbb{R}}^n)$ since $s\ge 1$, and F7 gives $Q\in W^{1,2}({\mathbb{R}}^n;\mathbb{C})$. Testing its weak derivative identity against the zero extension of any $\psi \in C_c^{\mathrm{\infty }}(G_0)$ gives $D_{i}w=(D_{i}Q){\mathrm{\mid }}_{G_0}$. Restriction decreases each $L^2$ component norm. More generally the same test calculation on any open $D$, followed by infimizing over all whole-space $H^1$ extensions of $h$, gives $\mathrm{\parallel }h{\mathrm{\parallel }}_{W^{1,2}(D)}\le C\mathrm{\parallel }h{\mathrm{\parallel }}_{H^1(D)}$. Thus $w\in W^{1,2}(G_0)$; F4 identifies its zero trace with the first-order trace in F5, and the bounded kernel theorem gives $w\in W_0^{1,2}(G_0;\mathbb{C})$. Choose ${\phi }_j\in C_c^{\mathrm{\infty }}(G_0;\mathbb{C})$ converging to $w$ in $W^{1,2}(G_0)$, using the closure definition and the countable-choice consequence of AC. By F6 their whole-space zero extensions satisfy $\mathrm{\parallel }{E}_0{\phi }_j-E_{0}w{\mathrm{\parallel }}_{W^{1,2}({\mathbb{R}}^n)}=\mathrm{\parallel }{\phi }_j-w{\mathrm{\parallel }}_{W^{1,2}(G_0)}\to 0$. F7 converts this to whole-space $H^1$ convergence. Their restrictions to $\mathrm{\Omega }$ lie in $C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ and converge in quotient $H^1(\mathrm{\Omega })$ by contractive restriction, hence also in $W^{1,2}(\mathrm{\Omega })$ by the bound just proved. The limit is $\theta U$: it agrees with $w$ on $G_0$ and both functions vanish on $\mathrm{\Omega }\setminus G_0$, since $\theta$ is supported inside $B_{L-3\epsilon }$. Thus the weak-norm closure definition gives $\theta U\in H_0^1(\mathrm{\Omega })$.

Take an ordinary whole-space $H^1$ extension $F$ of $U$ and set $W=(1-\theta )F$. F3 makes $W\in H^1({\mathbb{R}}^n)$, and F7 identifies it with a $W^{1,2}$ class. Choose a smooth bounded $b$ with bounded derivatives through order two, vanishing near $\mathrm{\Theta }$ and equal to $1$ on $\mathrm{supp}(1-\theta )$, so $bW=W$. By F5 choose $W_j\in C_c^{\mathrm{\infty }}({\mathbb{R}}^n;\mathbb{C})$ converging to $W$ in $W^{1,2}({\mathbb{R}}^n)$, using the countable-choice consequence of F1. F7 gives convergence in whole-space $H^1$; the fixed multiplier bound in F3 and contractive restriction give $(b{W}_j){\mathrm{\mid }}_{\mathrm{\Omega }}\to (1-\theta )U$ in $H^1(\mathrm{\Omega })$. The restriction bound in step 4.1 also gives convergence in $W^{1,2}(\mathrm{\Omega })$. Each $(b{W}_j){\mathrm{\mid }}_{\mathrm{\Omega }}$ lies in $C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ because $b$ vanishes on a fixed neighborhood of the compact obstacle. Hence the weak-norm closure definition gives $(1-\theta )U\in H_0^1(\mathrm{\Omega })$, and adding $\theta U$ proves the final assertion. This argument proves only the claimed integer zero-trace preservation and ordinary $H^s$ gluing; it does not impose any additional higher normal trace condition.