Collar traces and extension across an artificial boundary

Assume AC, let $n\ge 2$, and let $1\le s\le 2$. Throughout, use complex Bessel-potential Sobolev spaces: for $d\ge 1$ and $0\le t\le 2$, $H^t({\mathbb{R}}^d)$ consists of tempered distributions $U$ whose distributional product $(1+\mathrm{\mid }\xi {\mathrm{\mid }}^2{)}^{t\mathrm{/}2}\widehat{U}$ is represented by an $L^2$ function, with norm equal to the $L^2$ norm of that function. The Fourier transform has negative sign and $2\pi$ normalization. For an open set $D\subset {\mathbb{R}}^d$, use the restriction space and quotient norm $H^t(D)=\{U{\mathrm{\mid }}_D:U\in H^t({\mathbb{R}}^d)\},\mathrm{\parallel }u{\mathrm{\parallel }}_{H^t(D)}={\inf }_{U{\mathrm{\mid }}_D=u}\mathrm{\parallel }U{\mathrm{\parallel }}_{H^t({\mathbb{R}}^d)}\mathrm{.}$

For a compact smooth embedded hypersurface $S\subset {\mathbb{R}}^n$ without boundary, equip $S$ with its induced smooth structure and define $H^r(S)$, $1\mathrm{/}2\le r\le 3\mathrm{/}2$, by a finite compatible smooth atlas ${\kappa }_i:U_i\to V_i\subset {\mathbb{R}}^{n-1}$ and a subordinate smooth partition ${\chi }_i$ supported compactly inside the chart domains. For $f\in L^2(S)$ let $f_i=({\chi }_{i}f)\circ {\kappa }_i^{-1}$, extended by zero outside $V_i$, and set $\mathrm{\parallel }f{\mathrm{\parallel }}_{H^r(S)}^2={\sum }_i\mathrm{\parallel }{f}_i{\mathrm{\parallel }}_{H^r({\mathbb{R}}^{n-1})}^2$. These norms are independent of the chosen compatible finite atlas and partition up to equivalence. At order $3\mathrm{/}2$ this squared norm is equivalent to ${\sum }_i(\mathrm{\parallel }{f}_i{\mathrm{\parallel }}_{H^{1\mathrm{/}2}}^2+{\sum }_{a=1}^{n-1}\mathrm{\parallel }{\mathrm{\partial }}_a{f}_i{\mathrm{\parallel }}_{H^{1\mathrm{/}2}}^2)$. Thus the space is equivalently described by the lower-order norm and order-$1\mathrm{/}2$ first tangential weak derivatives of localized representatives, without choosing a global tangent frame.

Let $G\subset {\mathbb{R}}^n$ be a bounded smooth domain and let $S$ be a smooth boundary component. Smooth restriction extends to a bounded linear trace ${\gamma }_S:H^s(G)\to H^{s-1\mathrm{/}2}(S)$, agreeing in $L^2(S)$ with the first-order trace. For the interior version, let $U\subset {\mathbb{R}}^n$ be open, let $S\subset U$ be a compact smooth embedded hypersurface without boundary, and fix a bounded smooth collar/neighborhood $W⋐U$ containing $S$. Restriction to $W$ followed by its interior trace gives a bounded linear map $H^s(U)\to H^{s-1\mathrm{/}2}(S)$. The constants may depend on this fixed collar and its finite atlas. The traces are local, commute with smooth cutoffs, and are independent of extensions and local sides.

For the extension assertions, let $\mathrm{\Theta }\subset {\mathbb{R}}^n$ be a compact smooth obstacle, with connected smooth exterior $\mathrm{\Omega }={\mathbb{R}}^n\setminus \mathrm{\Theta }$. Write $B_L=\{x\in {\mathbb{R}}^n:\mathrm{\mid }x\mathrm{\mid }<L\}$, choose $L>0$ with $\mathrm{\Theta }⋐B_L$, and put $G_0=\mathrm{\Omega }\cap B_L$; its artificial boundary is $\mathrm{\partial }{B}_L$. If $v\in H^s(G_0)$ vanishes on a neighborhood of $\mathrm{\partial }{B}_L$ inside $G_0$, define $\widehat{v}=v$ on $G_0$ and $\widehat{v}=0$ on $\mathrm{\Omega }\setminus G_0$. Then $\widehat{v}\in H^s(\mathrm{\Omega })$. For each fixed outer gap and separating cutoff this partial zero extension is bounded, with constants allowed to depend on that gap/cutoff. No vanishing near $\mathrm{\partial }\mathrm{\Theta }$ is required, and no uniform bound for shrinking gaps is asserted. Here the hat on $v$ denotes partial zero extension, whereas the hat on $U$ in the Fourier convention above denotes Fourier transformation.

For bounded gluing, fix $0<\epsilon <L\mathrm{/}3$ with $\mathrm{\Theta }\subset B_{L-3\epsilon }$. Fix smooth cutoffs $\zeta ,\tilde{\zeta }$ with bounded derivatives, with $\zeta =0$ on $B_{L-2\epsilon }$, $\zeta =1$ for $\mathrm{\mid }x\mathrm{\mid }\ge L-\epsilon$, $\tilde{\zeta }=0$ on $B_{L-3\epsilon }$, and $\tilde{\zeta }=1$ on $\mathrm{supp}\zeta$. Put $\mathcal{A}=\{L-3\epsilon <\mathrm{\mid }x\mathrm{\mid }<L\}$ and fix one simultaneous bounded linear extension $E_{\mathcal{A}}:H^t(\mathcal{A})\to H^t({\mathbb{R}}^n)$, $0\le t\le 2$, agreeing with its input on $\mathcal{A}$. Then $\mathcal{G}u=\widehat{(1-\zeta )u}+{\tilde{\zeta }{E}_{\mathcal{A}}((\zeta u){\mathrm{\mid }}_{\mathcal{A}})\mid }_{\mathrm{\Omega }}$ is a bounded linear map $H^s(G_0)\to H^s(\mathrm{\Omega })$ with $(\mathcal{G}u){\mathrm{\mid }}_{G_0}=u$ and ${\gamma }_{\mathrm{\partial }\mathrm{\Theta }}\mathcal{G}u={\gamma }_{\mathrm{\partial }\mathrm{\Theta }}u$. In particular it preserves zero obstacle trace; it does not zero-extend across the obstacle. All gluing data are fixed.

The restriction-quotient convention and simultaneous extensions are supplied by the Sobolev extension/retraction result, the whole-space Fourier model by the Fourier characterization, and the intrinsic hypersurface norm by the atlas lemma. The collar trace and artificial-sphere gluing lemmas supply the stated trace and extension conclusions for the specified geometry. Constants retain their fixed collar, atlas, gap and cutoff dependence.