Exterior finite-energy functions have a trace controlled by their gradient

We justify the Sobolev slices. On each bounded spherical shell $a<\mathrm{\mid }x\mathrm{\mid }<b$ with $R\mathrm{/}2<a<R<b$, $u$ belongs to $H^1$: bounded volume and $L^6$ give $L^2$, and its weak gradient is already $L^2$. Use [F4] to approximate it by ambient smooth $u_m$ in $H^1$ on a slightly larger shell. The polar formula [F90] and $a^2\le r^2\le b^2$ bound ${\int }_{S^2}\mathrm{\parallel }{u}_m(r\omega )-u_l(r\omega ){\mathrm{\parallel }}_{H^1(a,b)}^{2}d\sigma (\omega )$ by $C\mathrm{\parallel }{u}_m-u_l{\mathrm{\parallel }}_{H^1}^2$, because the radial derivative of a smooth approximant is $\omega \cdot \mathrm{\nabla }{u}_m$. Take a subsequence with summable successive $H^1$ errors; Cauchy–Schwarz on the finite angular measure and Tonelli show that the successive slice norms are summable for almost every $\omega$. Completeness [F5] gives an $H^1(a,b)$ slice limit, identified by the polar $L^2$ limit with $u(r\omega )$ and derivative $\omega \cdot \mathrm{\nabla }u(r\omega )$. Countably many overlapping shells exhaust $r>R\mathrm{/}2$, so their representatives agree on overlaps off one angular null set; their values at $R$ define the usual trace, since the one-dimensional $H^1$ endpoint estimate passes smooth approximants to that value. By [F5] the slices are locally absolutely continuous.

For almost every direction $\omega \in S^2$, the radial slice $r\mapsto u(r\omega )$ is locally H1 for $r\ge R$, and ${\int }_R^{\mathrm{\infty }}\mathrm{\mid }u(r\omega ){\mathrm{\mid }}^6{r}^{2}dr<\mathrm{\infty }$. Thus there is a sequence $r_m\to \mathrm{\infty }$ with $u(r_m\omega )\to 0$. For each such direction the radial derivative is square integrable with weight $r^2$; the one-dimensional fundamental theorem on $[R,r_m]$ and Cauchy–Schwarz give $\mathrm{\mid }u(R\omega ){\mathrm{\mid }}^2\le ({\int }_R^{\mathrm{\infty }}\mathrm{\mid }{\mathrm{\partial }}_{r}u(r\omega ){\mathrm{\mid }}^2{r}^{2}dr)\left({\int }_R^{\mathrm{\infty }}{r}^{-2}dr\right)=R^{-1}{\int }_R^{\mathrm{\infty }}\mathrm{\mid }{\mathrm{\partial }}_{r}u(r\omega ){\mathrm{\mid }}^2{r}^{2}dr\mathrm{.}$ Integrating over $S^2$ by [F3,F90] yields $\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2(\mathrm{\partial }{B}_R)}\le R^{1\mathrm{/}2}\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_{L^2({\mathrm{\Omega }}^{+}\setminus B_R)}$.

The bounded annular domain $A={\mathrm{\Omega }}^{+}\cap B_R$ is Lipschitz: its boundary components are the original Lipschitz $\mathrm{\Gamma }$ and the separated smooth sphere. Hence [F4] gives the exact extension-domain property required by [F1]. It is connected: for any point inside $B_{R\mathrm{/}2}$, path-connectivity of the unbounded open set ${\mathrm{\Omega }}^{+}$ gives a path to a point outside $B_{R\mathrm{/}2}$; stopping at the first crossing of $\mathrm{\partial }{B}_{R\mathrm{/}2}$ keeps the path in $A$. The sphere $\mathrm{\partial }{B}_{R\mathrm{/}2}$ lies wholly in $A$ and is connected, and the outer shell $R\mathrm{/}2<\mathrm{\mid }x\mathrm{\mid }<R$ is connected, so all points of $A$ join within $A$. Put $m=\mathrm{\mid }A{\mathrm{\mid }}^{-1}{\int }_{A}u$. By [F1], $\mathrm{\parallel }u-m{\mathrm{\parallel }}_{L^2(A)}\le C\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_{L^2(A)}$. A direct one-dimensional trace estimate in the radial collar $R-\delta <\mathrm{\mid }x\mathrm{\mid }<R$ gives $\mathrm{\parallel }u-m{\mathrm{\parallel }}_{L^2(\mathrm{\partial }{B}_R)}\le C\mathrm{\parallel }u-m{\mathrm{\parallel }}_{H^1(A)}\le C\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_{L^2(A)}$: for each ray apply $\mathrm{\mid }w(R){\mathrm{\mid }}^2\le C_{\delta }{\int }_{R-\delta }^R(\mathrm{\mid }w(r){\mathrm{\mid }}^2+\mathrm{\mid }{w}^{\mathrm{\prime }}(r){\mathrm{\mid }}^2)dr$ to $w(r)=u(r\omega )-m$, then integrate using that $r^2$ is comparable to $R^2$ on the collar. Hence $\mathrm{\mid }m\mathrm{\mid }\mathrm{\mid }\mathrm{\partial }{B}_R{\mathrm{\mid }}^{1\mathrm{/}2}\le \mathrm{\parallel }u{\mathrm{\parallel }}_{L^2(\mathrm{\partial }{B}_R)}+\mathrm{\parallel }u-m{\mathrm{\parallel }}_{L^2(\mathrm{\partial }{B}_R)}\le C\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_{L^2({\mathrm{\Omega }}^{+})}$ by step 2.1. It follows that $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^1(A)}\le C\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_{L^2({\mathrm{\Omega }}^{+})}$.

The annular estimate of step 3.1 controls the H1 norm on an exterior collar of $\mathrm{\Gamma }$. Apply [F2] to obtain $\mathrm{\parallel }{\gamma }^{+}u{\mathrm{\parallel }}_{H^{1\mathrm{/}2}(\mathrm{\Gamma })}\le C\mathrm{\parallel }u{\mathrm{\parallel }}_{H^1(A)}\le C\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_{L^2({\mathrm{\Omega }}^{+})}$.