Collar and tail assembly of the exterior graph domain

Choose $L>0$ with $\mathrm{\Theta }⋐B_L$ and, by [F6] with compact set $\mathrm{\Theta }\subset B_L$, choose $\eta \in C_c^{\mathrm{\infty }}(B_L)$ equal to $1$ near $\mathrm{\Theta }$. Put $G=B_L\setminus \mathrm{\Theta }$, exactly as in [F3]. Let $u$ belong to the weak domain and put $g=\mathrm{\Delta }u$. The collar lemma gives $w=\eta u\in H^2(G)$ and $\mathrm{\parallel }w{\mathrm{\parallel }}_{H^2(G)}\le C(\mathrm{\parallel }g{\mathrm{\parallel }}_2+\mathrm{\parallel }u{\mathrm{\parallel }}_{H^1})$. Define $\tilde{w}=w$ on $G$ and zero on $\mathrm{\Omega }\setminus G$. The two open sets $G$ and $\mathrm{\Omega }\setminus \mathrm{supp}\eta$ cover $\mathrm{\Omega }$. On the second set $\tilde{w}$ is identically zero, and on the overlap $w$ and all its weak derivatives through order two vanish. For each compactly supported test on $\mathrm{\Omega }$, a subordinate smooth partition from [F6] has only finitely many terms meeting its support. Applying the weak derivative identities on each open piece and summing proves the global weak identities; thus $\tilde{w}\in H^2(\mathrm{\Omega })$, with each derivative given by zero extension of the corresponding derivative on $G$ and with unchanged component norms. This extends only across the artificial sphere, not across the obstacle.

The tail lemma and whole-space Fourier lemma give a function $v\in H^2({\mathbb{R}}^n)$ with $v{\mathrm{\mid }}_{\mathrm{\Omega }}=(1-\eta )u$ and $\mathrm{\parallel }v{\mathrm{\parallel }}_{H^2}\le C(\mathrm{\parallel }g{\mathrm{\parallel }}_2+\mathrm{\parallel }u{\mathrm{\parallel }}_{H^1})$. Adding the extended collar part to $v{\mathrm{\mid }}_{\mathrm{\Omega }}$ gives $u\in H^2(\mathrm{\Omega })$ and $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^2}\le C(\mathrm{\parallel }g{\mathrm{\parallel }}_2+\mathrm{\parallel }u{\mathrm{\parallel }}_{H^1})$. Conversely $u\in H^2\cap H_0^1$ has $\mathrm{\Delta }u\in L^2$ by its weak derivatives, so the two domains coincide.

For $u\in D(A)$, the weak form bound gives $\mathrm{\parallel }u{\mathrm{\parallel }}_2\le \mathrm{\parallel }u{\mathrm{\parallel }}_{H^1}\le \mathrm{\parallel }Au{\mathrm{\parallel }}_2$, while $g=u-Au$ gives $\mathrm{\parallel }g{\mathrm{\parallel }}_2\le 2\mathrm{\parallel }Au{\mathrm{\parallel }}_2$. Substitution in the preceding estimate proves $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^2}\le C\mathrm{\parallel }Au{\mathrm{\parallel }}_2$. In the other direction $\mathrm{\parallel }Au{\mathrm{\parallel }}_2=\mathrm{\parallel }u-\mathrm{\Delta }u{\mathrm{\parallel }}_2\le C_n\mathrm{\parallel }u{\mathrm{\parallel }}_{H^2}$. These are the stated graph-norm comparisons.