Trace formula and bounded zero extension on Lipschitz domains

If $\mathrm{\Omega }=\mathrm{\varnothing }$, its BV space and boundary $L^1$ space are both the zero space, the zero trace is unique, and every formula and estimate holds with both sides zero. Assume henceforth $\mathrm{\Omega }\mathrm{\ne }\mathrm{\varnothing }$. By [F5], $\mathrm{\partial }\mathrm{\Omega }$ is compact; choose finitely many smaller graph charts with positive margins inside the original charts that still cover it. Each smaller graph base is bounded and its Lipschitz constant finite, so [F2] gives finite Radon surface measure on that patch and ${\mathcal{H}}^{n-1}(\mathrm{\partial }\mathrm{\Omega })<\mathrm{\infty }$. In each chart [F1] defines a local trace and bounds its surface $L^1$ norm, using [F2], by a chart constant times $\mathrm{\parallel }u{\mathrm{\parallel }}_1+\mathrm{\mid }Du\mathrm{\mid }(\mathrm{\Omega })$. On overlaps these local traces agree by [F1]. Consequently they define one class $Tu\in L^1(\mathrm{\partial }\mathrm{\Omega },{\mathcal{H}}^{n-1})$ with $\mathrm{\parallel }Tu{\mathrm{\parallel }}_1\le C(\mathrm{\parallel }u{\mathrm{\parallel }}_1+TV(u;\mathrm{\Omega }))$. Linearity follows locally from [F1], hence globally.

Construct a finite smooth partition equal to one on a neighborhood of $\bar{\mathrm{\Omega }}$, subordinate to the boundary charts and the interior open set $\mathrm{\Omega }$: by [F5] this compact set is covered by finitely many balls with doubled closures in those open sets; take the bumps in [F3], and divide them by their positive sum near $\bar{\mathrm{\Omega }}$. Multiply by one further smooth cutoff supported where the sum is positive and equal to one near $\bar{\mathrm{\Omega }}$. First take $\varphi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^n;{\mathbb{R}}^n)$ and apply the local Green identity [F1] to each chart-supported field ${\eta }_j\varphi$. For the interior-supported field the defining distributional identity applies. Sum the identities. Since ${\sum }_j{\eta }_j=1$ and ${\sum }_j\mathrm{\nabla }{\eta }_j=0$ near $\bar{\mathrm{\Omega }}$, the interior and boundary sums are exactly ${\int }_{\mathrm{\Omega }}u\mathrm{div}\varphi =-{\int }_{\mathrm{\Omega }}\varphi \cdot dDu+{\int }_{\mathrm{\partial }\mathrm{\Omega }}Tu\varphi \cdot \nu d{\mathcal{H}}^{n-1}\mathrm{.}$ For an arbitrary $\varphi \in C_c^1$, use [F7] to approximate $\varphi$ and its first derivatives uniformly by compactly supported smooth fields; finite variation and $Tu\in L^1$ pass every term to the limit.

Extend each coordinate of $Du$ by zero from $\mathrm{\Omega }$ to ${\mathbb{R}}^n$. It is a finite signed Radon measure: compact inner approximation can be performed inside $\mathrm{\Omega }$, and outer approximation in the ambient space follows by applying that inner approximation to the complement and using finiteness. For each coordinate $i$, the boundary density $f_i:=Tu{\nu }_i$ is integrable because $Tu\in L^1$ and $\mathrm{\mid }{\nu }_i\mathrm{\mid }\le 1$. By [F6], $f_i{\mathcal{H}}^{n-1}{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}$ is a finite signed Radon measure; this verifies the required weighted-measure regularity, not just its finiteness. Thus their difference $\mu =i_{\mathrm{\#}}Du-Tu\nu {\mathcal{H}}^{n-1}{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}$ is a finite Radon vector measure by [F4]. The formula in step 2.1 says exactly that $D\tilde{u}=\mu$, so $\tilde{u}\in BV({\mathbb{R}}^n)$.

The two summands defining $\mu$ in step 3.1 are supported on the disjoint Borel sets $\mathrm{\Omega }$ and $\mathrm{\partial }\mathrm{\Omega }$. The partition definition of variation therefore gives the sum of their variations. For a unit measurable vector density $\nu$ and scalar density $Tu$, the boundary variation is $\mathrm{\mid }Tu\mathrm{\mid }{\mathcal{H}}^{n-1}$: the triangle inequality gives the upper bound, and a finite partition of the unit sphere into pieces of diameter below $ϵ$, applied to the direction $\mathrm{sgn}(Tu)\nu$, gives the lower bound $(1-ϵ)\mathrm{\mid }Tu\mathrm{\mid }{\mathcal{H}}^{n-1}$ by projection; let $ϵ\downarrow 0$. Hence $TV(\tilde{u})=TV(u;\mathrm{\Omega })+\mathrm{\parallel }Tu{\mathrm{\parallel }}_1$. Step 1.1 gives the asserted bound. If another trace satisfied the Green identity, their difference times $\nu {\mathcal{H}}^{n-1}$ would integrate to zero on all smooth tests, hence be zero by [F3]; the same variation calculation gives equality of the trace classes.

In dimension one, each smaller graph neighborhood contains one boundary point, so the finite graph cover makes the boundary finite. A bounded open set with finite boundary is a finite disjoint union of bounded intervals. Use the one-sided endpoint construction in [F1] on each of their finitely many endpoints. The partition argument of step 2.1 and the disjoint interior/boundary measure calculation in steps 3.1–4.1 apply verbatim with ${\mathcal{H}}^0$ counting measure and outward signs $-1$ at a left endpoint and $+1$ at a right endpoint. This proves the stated dimension-one companion.