Exterior spectral domains and the critical trace index

Choose $L$ and smooth cutoffs $\eta ,a,\beta$ as in the collar–tail supplier: $\mathrm{\Theta }⋐B_{L\mathrm{/}4}$, $\eta =1$ near $\mathrm{\Theta }$ and is supported in $B_{L\mathrm{/}2}$, $a=1$ near $\mathrm{supp}\eta$ and is supported in $B_L$, and $\beta =1$ near $\mathrm{\Theta }$ with support in the interior of the plateau $\{\eta =1\}$. Put $b=1-\beta$, $G=\mathrm{\Omega }\cap B_L$, and let $Zu$ be the zero extension to ${\mathbb{R}}^n$ of $(1-\eta )u$. Define $Ju=((\eta u){\mathrm{\mid }}_G,Zu)$ and $P(v,w)=V_{a}v+(bw){\mathrm{\mid }}_{\mathrm{\Omega }}$, where $V_{a}v$ is $av$ on $G$ and zero on $\mathrm{\Omega }\setminus G$. The supplier proves both maps bounded on the ordinary $L^2$ and $H^2$ endpoint couples, and $PJ=I$. In particular the unbounded tail zero extension is justified there by compactly supported tests, and $V_a$ crosses only the artificial sphere, where $a$ vanishes.

If $u\in H^2(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega })$, choose $u_j\in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ converging to $u$ in $H^1(\mathrm{\Omega })$. The functions $(\eta u_j){\mathrm{\mid }}_G$ lie in $C_c^{\mathrm{\infty }}(G)$ and converge to $(\eta u){\mathrm{\mid }}_G$ in $H^1(G)$ by the product rule. Thus the first component of $Ju$ lies in $H^2(G)\cap H_0^1(G)$; the second lies in $H^2({\mathbb{R}}^n)$. Conversely, the collar–tail supplier proves that $P$ sends the product of those two endpoint spaces into $H^2(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega })$. Hence the same $J,P$ form a bounded retraction pair for the Dirichlet endpoint couples as well as for $L^2$.

Let $0<s<2$. Apply the strip-map theorem to those two common endpoint maps. Coordinate projections and injections give one direction of the finite-product interpolation norm comparison; pairing admissible component strip functions gives the other. Consequently $J$ and $P$ give a retraction between $[L^2(\mathrm{\Omega }),H^2(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega }){]}_{s\mathrm{/}2}$ and the product of the bounded Dirichlet interpolation space on $G$ with $H^s({\mathbb{R}}^n)$, with equivalent norms. By the spectral scale this exterior interpolation space is $H_D^s(\mathrm{\Omega })$. Thus $u\in H_D^s$ exactly when its collar component belongs to the bounded Dirichlet interpolation space and $Zu\in H^s({\mathbb{R}}^n)$; its scale norm is equivalent to the sum of those component norms.

The ordinary collar–tail retraction says that $u\in H^s(\mathrm{\Omega })$ exactly when $(\eta u){\mathrm{\mid }}_G\in H^s(G)$ and $Zu\in H^s({\mathbb{R}}^n)$, with equivalent norms. The bounded-domain input therefore gives the claimed result for $0<s<1\mathrm{/}2$. For $1\mathrm{/}2<s<2$, define ${\gamma }_{0}u$ as the trace of $(\eta u){\mathrm{\mid }}_G$ on the obstacle boundary. Its trace on the artificial sphere is zero because $\eta$ vanishes in a neighborhood of that sphere. Since $\eta =1$ near the obstacle, the obstacle trace is independent of the chosen such cutoff: the difference has support away from that boundary and has zero localized trace. The bounded-domain input now gives precisely ${\gamma }_{0}u=0$ and no further condition, including at $s=3\mathrm{/}2$.

At $s=1\mathrm{/}2$, the bounded input requires $d_G^{-1\mathrm{/}2}(\eta u){\mathrm{\mid }}_G\in L^2(G)$, where $d_G=\mathrm{dist}(\cdot ,\mathrm{\partial }G)$. In a fixed neighborhood of the obstacle, $d_G=d$ after choosing the neighborhood sufficiently small; away from that neighborhood, $d$ is bounded below, while $\eta$ vanishes near the artificial sphere and $d_G$ is bounded below on the remaining relevant support. Also $d^{-1}$ is bounded on the exterior tail. Splitting the integrals into these regions shows that the bounded collar weighted norm plus the tail $H^{1\mathrm{/}2}$ norm is equivalent to $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^{1\mathrm{/}2}(\mathrm{\Omega })}+\mathrm{\parallel }{d}^{-1\mathrm{/}2}u{\mathrm{\parallel }}_2$. This proves the critical assertion without identifying it with the $H^{1\mathrm{/}2}$-closure of test functions.

At $s=0$, $H_D^0=L^2=H^0$; at $s=2$, the realization theorem gives $H_D^2=H^2\cap H_0^1$ with equivalent norms. For $u\in H^1(\mathrm{\Omega })$, zero obstacle trace implies $(\eta u){\mathrm{\mid }}_G\in H_0^1(G)$ by the bounded trace-kernel theorem, since its artificial-sphere trace is zero. Approximate that collar component by compactly supported smooth functions in $G$ and approximate $Zu\in H^1({\mathbb{R}}^n)$ by compactly supported smooth whole-space functions. Applying $P$ to these approximants yields compactly supported smooth functions in $\mathrm{\Omega }$ converging to $u$, because $a$ is supported inside $B_L$ and $b$ vanishes near the obstacle. Conversely, restriction and smooth multiplication carry any $H_0^1(\mathrm{\Omega })$ approximating sequence to a test-function approximating sequence for the collar component, so its trace vanishes. Thus $\ker {\gamma }_0=H_0^1(\mathrm{\Omega })$ in $H^1(\mathrm{\Omega })$. The $H^s$-to-$H^1$ inclusion for $s\ge 1$ follows from the restriction quotient and the whole-space Fourier weights. Combining this with the previous steps proves the stated $1\le s\le 2$ reformulation and all endpoint claims.