One-sided BV traces and Green identity on a Lipschitz graph

Fix $a>0$ with $C_{2a}$ inside the larger chart. On every compact subset of $G$, mollify $u$ using a nonnegative unit-mass bump, writing $u_{ϵ}=u\ast {\rho }_{ϵ}$ where the convolution ball stays in $G$. The distributional derivative identity and [F2] give ${\mathrm{\partial }}_z{u}_{ϵ}=(D_{z}u)\ast {\rho }_{ϵ}$ there. For $0<s<t<a$, ordinary line integration gives ${\int }_B\mathrm{\mid }{u}_{ϵ}(y,h(y)+t)-u_{ϵ}(y,h(y)+s)\mathrm{\mid }dy\le {\int }_{\{y\in B,s<z-h(y)<t\}}\mathrm{\mid }{\mathrm{\partial }}_z{u}_{ϵ}\mathrm{\mid }dx\mathrm{.}$ For sufficiently small $ϵ$, the $ϵ$-neighborhood of this last strip lies in $C_{2a}$: its lower height is positive since $s>0$, its upper height is below $2a$, and its lateral projection lies in $B^{\mathrm{\prime }}$. The convolution bound and [F2] make the right side at most $\mathrm{\mid }{D}_{z}u\mathrm{\mid }(C_{2a})$, since each kernel integrates to at most one.

On each compact-height collar $y\in B$, $1\mathrm{/}j<z-h(y)<a-1\mathrm{/}j$, [F1] gives $L^1$ convergence of $u_{ϵ}$ to $u$. Choose a sequence ${ϵ}_{\mathrm{\ell }}\downarrow 0$ whose $L^1$ errors on the first $\mathrm{\ell }$ such collars are below $2^{-\mathrm{\ell }}$, always using the least sufficiently large dyadic exponent. By [F2], for a full-measure set of heights the slice functions converge in $L^1(B)$. Applying step 1.1 at two such heights proves $\mathrm{\parallel }{u}_t-u_s{\mathrm{\parallel }}_{L^1(B)}\le \mathrm{\mid }{D}_{z}u\mathrm{\mid }(C_{2a}),u_t(y)=u(y,h(y)+t),0<s,t<a\mathrm{.}$ Repeat this construction for a countable sequence of smaller $a$ tending to zero and intersect the full-measure height sets. The collars $C_{2a}$ decrease to the empty set inside $G$, so finiteness of $\mathrm{\mid }{D}_{z}u\mathrm{\mid }$ implies their measures tend to zero. Thus $u_t$ is Cauchy in $L^1(B)$ as $t\downarrow 0$ outside one null set.

Choose one height from each interval $(2^{-j-1}a,2^{-j}a)$ outside that null set by Countable Choice. The corresponding Cauchy sequence has an $L^1$ limit: choose measurable representatives, then a subsequence whose successive $L^1$ differences are below $2^{-j}$; [F2] makes the sum of the absolute differences integrable and gives its almost-everywhere and $L^1$ limit. The Cauchy estimate in step 2.1 makes this limit independent of heights and gives the asserted convergence along all good heights. This also proves uniqueness and linearity. Averaging the bound $\mathrm{\parallel }\tau u{\mathrm{\parallel }}_1\le \mathrm{\parallel }{u}_t{\mathrm{\parallel }}_1+\mathrm{\mid }{D}_{z}u\mathrm{\mid }(C_{2a})$ over $0<t<a$ using [F2] gives $\mathrm{\parallel }\tau u{\mathrm{\parallel }}_1\le a^{-1}\mathrm{\parallel }u{\mathrm{\parallel }}_{L^1(C_a)}+\mathrm{\mid }{D}_{z}u\mathrm{\mid }(C_{2a})$. Since $\mathrm{\mid }{D}_{z}u\mathrm{\mid }\le \mathrm{\mid }Du\mathrm{\mid }$, this is the stated bound. Traces obtained on nested base boxes coincide by the same slice convergence.

The distributional identity extends from smooth to compactly supported Lipschitz vector fields $\psi$ inside $G$. Indeed mollification converges uniformly to $\psi$, while its derivatives are uniformly bounded and converge almost everywhere to the weak derivatives, by [F3] and Lebesgue differentiation: if $g$ is a bounded weak derivative at a strong Lebesgue point $x$, then $\mathrm{\mid }g\ast {\rho }_{ϵ}(x)-g(x)\mathrm{\mid }\le C{ϵ}^{-n}{\int }_{B(x,ϵ)}\mathrm{\mid }g(y)-g(x)\mathrm{\mid }dy\to 0$. Consequently $\int {\psi }_{ϵ}\cdot dDu\to \int \psi \cdot dDu$ by uniform convergence and finiteness, and $\int u\mathrm{div}{\psi }_{ϵ}\to \int u\mathrm{div}\psi$ by dominated convergence. Set $r=z-h(y)$ and let $\theta$ be the continuous function zero on $(-\mathrm{\infty },0]$, linear from zero to one on $[0,1]$, and one on $[1,\mathrm{\infty })$. For fixed $ϵ>0$ use $\psi =\varphi \theta ((r-ϵ)\mathrm{/}ϵ)$, whose support is a compact subset of $G$. Its almost-everywhere gradient uses the ordinary chain rule from [F3], and its transition term is ${ϵ}^{-1}{\int }_{ϵ}^{2ϵ}\int u_t(y)\varphi (y,h(y)+t)\cdot (-Dh(y),1)dydt\mathrm{.}$ By step 3.1, the uniform bound on $Dh$, and uniform continuity of $\varphi$, this converges to $\int \tau u\varphi (y,h(y))\cdot (-Dh,1)dy$. The other terms converge to ${\int }_{G}u\mathrm{div}\varphi$ and $-{\int }_G\varphi \cdot dDu$, since the cutoff tends pointwise to one in $G$. Rearrangement proves the Green formula.

These formulas also show independence of the direction of a valid graph chart. On an overlap, [F3] applied to $u=1$ first identifies the outward vector surface measures by [F4]. Their variations are their scalar graph measures: for a unit measurable density $n$ against a finite positive measure $\mu$, variation is at most $\mu$ by the triangle inequality, and is at least $(1-\eta )\mu$ by partitioning the unit sphere into finitely many Borel sets of diameter below $\eta$ and projecting each vector integral onto a representative direction. Let $\eta \downarrow 0$. Thus the outward normals agree almost everywhere. Subtract the two Green formulas for general $u$; [F4] makes $({\tau }_{1}u-{\tau }_{2}u)n\mu$ zero, and the same variation argument gives ${\tau }_{1}u={\tau }_{2}u$ almost everywhere. Reflection gives the subgraph formulas. In dimension one, omit the base integral: the convolution and line estimate in steps 1.1–2.1 make $u(t)$ Cauchy through almost every $t$ approaching an endpoint, and step 4.1 gives the endpoint boundary term. All arguments and bounds remain valid, with scalar absolute value and counting measure.