H1 traces agree across interior mesh facets

Fix a compact $(d-1)$-simplex $G$ contained in the relative interior of $F$. A sufficiently small open tube around $G$ has closure in $D$. Cover its closure by finitely many balls whose larger concentric balls still have closure in $D$; [F2] supplies a smooth cutoff equal to one on each smaller ball and supported in its larger ball. The function $\chi =1-{\prod }_i(1-{\chi }_i)$ is compactly supported in $D$ and equals one near $G$. The zero extension $W$ of $\chi w$ belongs to $H^1({\mathbb{R}}^d)$: for a test function $\psi$, the weak product rule follows directly from ${\int }_D\chi w{\mathrm{\partial }}_i\psi ={\int }_{D}w{\mathrm{\partial }}_i(\chi \psi )-{\int }_{D}w({\mathrm{\partial }}_i\chi )\psi =-{\int }_D(\chi {\mathrm{\partial }}_{i}w+w{\mathrm{\partial }}_i\chi )\psi ,$ and compact support of $\chi$ removes any boundary term on extension.

Let $W_{\epsilon }={\rho }_{\epsilon }\ast W$ with the mollifier of [F2]. By [F3] applied to $W$ and its weak derivatives, and by derivative commutation on the tube where $\chi =1$, $W_{\epsilon }\to w$ in $H^1$ on every smaller tube around $G$. Each $W_{\epsilon }$ is smooth by [F4]. Cover $G$ by finitely many smaller simplices on each side whose shared facet portions lie in that tube. The trace bound [F1] on each smaller simplex shows the traces of $W_{\epsilon }$ converge in $L^2$ there to the local traces of $w$. By locality and uniqueness of the bounded trace extension, these are the restrictions of the elementwise traces defined on $K_{+}$ and $K_{-}$. The smooth $W_{\epsilon }$ has identical restrictions from both sides, so the two traces are equal; their limits are equal as well.

Countably many compact facet simplices $G$ exhaust the relative interior of $F$; its boundary has $(d-1)$-dimensional measure zero. Step 2.1 gives equality on each such patch, hence equality almost everywhere on all of $F$.