Whole-space support and the zero-boundary Sobolev closure

Localize $F$ by a finite smooth partition subordinate to an interior set and boundary graph charts. Rotate a chart so that $\mathrm{\Omega }$ is $x_3>\varphi (x^{\mathrm{\prime }})$. Its localized whole-space $H^1$ function is zero below this graph and has support away from the chart sides. Translate that localized function upward by $t{e}_3$. Its support now lies at least $t\mathrm{/}\sqrt{1+L^2}$ from the graph, because $\varphi$ is $L$-Lipschitz. Mollification with radius less than $t\mathrm{/}(2(1+L))$ produces a smooth function compactly supported in $\mathrm{\Omega }$. Translation and then sufficiently fine mollification converge in whole-space $H^1$ by [F1]. The localized interior term is mollified directly. Summing the finite approximations proves $F{\mathrm{\mid }}_{\mathrm{\Omega }}\in H_0^1$. Conversely a sequence of compactly supported smooth approximants in the closure definition has zero extensions Cauchy in $H^1({\mathbb{R}}^3)$, with exactly the original norm differences. The limit is the zero extension of its $L^2$ limit and proves the converse. Lipschitz boundaries have three-dimensional measure zero, since every graph has zero vertical section measure, so support in $\bar{\mathrm{\Omega }}$ causes no additional $L^2$ term.

For clause 2 localize $f$ in a graph chart and write its vertical height as $s=x_3-\varphi (x^{\mathrm{\prime }})>0$. The continuous boundary value is zero. For almost every $x^{\mathrm{\prime }}$, [F2] and continuity at the boundary give $\mathrm{\mid }f(x^{\mathrm{\prime }},\varphi (x^{\mathrm{\prime }})+s){\mathrm{\mid }}^2\le s{\int }_0^s\mathrm{\mid }{\mathrm{\partial }}_{3}f(x^{\mathrm{\prime }},\varphi (x^{\mathrm{\prime }})+r){\mathrm{\mid }}^{2}dr\mathrm{.}$ To justify the endpoint, first integrate the ACL identity between $\epsilon$ and $s$, then let $\epsilon \downarrow 0$; the $L^2$ derivative is integrable along almost every finite vertical segment, and continuity gives the endpoint limit. Choose a fixed smooth scalar $\eta$ that is zero on $(-\mathrm{\infty },1]$ and one on $[2,\mathrm{\infty })$, and multiply by $\eta (s\mathrm{/}t)$. The construction, Sobolev chain rule and bounded-Sobolev product rule in [F3]–[F4] justify this nonsmooth-in-space factor and bound its weak gradient by $C(1+L)\mathrm{/}t$. The norm of its extra gradient term is bounded by $C\mathrm{\parallel }\mathrm{\nabla }f{\mathrm{\parallel }}_{L^2(0<s<2t)}$, using the displayed inequality and integration in $s,x^{\mathrm{\prime }}$, and hence tends to zero. The removed function and gradient terms also tend to zero by absolute continuity of their integrals. These cutoffs lie away from the boundary and can be mollified inside $\mathrm{\Omega }$; compact chart supports handle their other sides. Sum the finite local approximations and the interior term. Thus $f$ is in the asserted closure.

Choose $u_j\in C_c^{\mathrm{\infty }}(\mathrm{\Omega };{\mathbb{R}}^3)$ converging to $u$ in the H(curl) graph norm. Their zero extensions are Cauchy in the whole-space graph norm. The weak test identity passes to the $L^2$ limits of the fields and their curls by Cauchy–Schwarz. Those limits are respectively $U$ and the zero extension of $\mathrm{curl}u$, proving the asserted distributional identity and both norm equalities. For $q\in H_0^1$, take $q_j\in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ converging in $H^1$; then $\mathrm{\nabla }{q}_j\to \mathrm{\nabla }q$ in $L^2$, their curls are zero by equality of smooth mixed derivatives, and $\mathrm{\nabla }{q}_j\in C_c^{\mathrm{\infty }}(\mathrm{\Omega };{\mathbb{R}}^3)$. The same weak-limit argument proves the last claim. AC licenses these closure approximation sequences and the cited analytic interfaces.