Sobolev face traces and Gauss–Green on a reference simplex

Fix a facet $F$ opposite the vertex $z_F$, let $H_F>0$ be the altitude from $z_F$ to $F$, and put $X(x)=x-z_F$. On $F$, $X\cdot {\widehat{n}}_F=H_F$; on every other facet, which contains $z_F$, $X$ is tangent, so its normal component is zero. For a smooth scalar $u$, apply [F2] to $\mathrm{\mid }u{\mathrm{\mid }}^{p}X$. Since $p>1$, $\mathrm{\mid }u{\mathrm{\mid }}^p$ is continuously differentiable, and $\mathrm{\mid }X\mathrm{\mid }\le \mathrm{diam}({\widehat{S}}_d)$; hence $H_F{\int }_F\mathrm{\mid }u{\mathrm{\mid }}^{p}dS\le d{\int }_{{\widehat{S}}_d}\mathrm{\mid }u{\mathrm{\mid }}^{p}dx+p\mathrm{diam}({\widehat{S}}_d){\int }_{{\widehat{S}}_d}\mathrm{\mid }u{\mathrm{\mid }}^{p-1}\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid }dx\mathrm{.}$ Holder bounds the last integral by $\mathrm{\parallel }u{\mathrm{\parallel }}_p^{p-1}\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_p$. For $a=\mathrm{\parallel }u{\mathrm{\parallel }}_p$ and $b=\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_p$, if $b\le a$ then $a^{p-1}b\le a^p$, while if $a\le b$ then $a^{p-1}b\le b^p$. Thus the product is at most $a^p+b^p$. Since $H_F>0$, the resulting inequality gives $\mathrm{\parallel }u{\mathrm{\parallel }}_{L^p(F)}\le C_{F,p}\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{1,p}({\widehat{S}}_d)}$.

Let $(u_j)$ be a smooth sequence converging to $u$ in $W^{1,p}({\widehat{S}}_d)$, which exists by [F1]. Applying the estimate of step 1.1 to $u_j-u_i$ shows $(u_j{\mathrm{\mid }}_F)$ is Cauchy in the complete space $L^p(F)$; define ${\gamma }_{F}u$ to be its limit. The same estimate shows the limit is independent of the chosen smooth approximating sequence and $\mathrm{\parallel }{\gamma }_{F}u{\mathrm{\parallel }}_{L^p(F)}\le C\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{1,p}}$. Linearity follows by taking approximating sequences for sums and scalar multiples.

For smooth $u,G$, apply [F2] to the vector field $uG$. Its divergence is $\mathrm{\nabla }u\cdot G+u\mathrm{div}G$, so the displayed Green identity holds with ordinary restrictions. Choose smooth approximations $u_j\to u$ in $W^{1,p}$ and $G_j\to G$ componentwise in $W^{1,p^{\mathrm{\prime }}}$ by [F1]. By step 2.1 their facet traces converge in $L^p$ and $L^{p^{\mathrm{\prime }}}$; Holder [F3] makes the boundary products converge in $L^1(F)$. The volume products converge in $L^1$ by Holder and convergence of both Sobolev norms. Passing to the limit in the smooth identity proves the formula for $u,G$.