H1 facet trace inequality on shape-regular simplices

It suffices first to prove the reference estimate for a smooth function. On the reference simplex choose a facet $\widehat{F}$ opposite a vertex $a$, and parameterize inward segments by $x(y,t)=(1-t)y+ta$, $y\in \widehat{F}$, $0\le t\le \frac{1}{2}$. For fixed $y$ and $t$ in $[\frac{1}{4},\frac{1}{2}]$, the fundamental theorem of calculus and Cauchy–Schwarz give $\mathrm{\mid }u(y){\mathrm{\mid }}^2\le 2\mathrm{\mid }u(x(y,t)){\mathrm{\mid }}^2+2t{\int }_0^t\mathrm{\mid }(a-y)\cdot \mathrm{\nabla }u(x(y,s)){\mathrm{\mid }}^2,ds\mathrm{.}$ Integrate in $y$ and $t$. The segment Jacobian is a fixed positive constant times $(1-t{)}^{d-1}$, bounded above and below on this interval; $\mathrm{\mid }a-y\mathrm{\mid }$ is bounded on the compact reference simplex. Changing variables in the first integral and using Fubini [F4] in the second bounds $\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2(\widehat{F})}^2$ by $C_d(\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2(\widehat{K})}^2+\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_{L^2(\widehat{K})}^2)$.

For a general affine simplex write $x=b+A\widehat{x}$ and $\widehat{u}=u\circ F$. The affine change of variables [F5] gives $\mathrm{\parallel }\widehat{u}{\mathrm{\parallel }}_{L^2(\widehat{K})}^2=\mathrm{\mid }\det A{\mathrm{\mid }}^{-1}\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2(K)}^2$ and $\mathrm{\parallel }{\mathrm{\nabla }}_{\widehat{x}}\widehat{u}{\mathrm{\parallel }}_2^2\le \mathrm{\mid }\det A{\mathrm{\mid }}^{-1}\mathrm{\parallel }A{\mathrm{\parallel }}^2\mathrm{\parallel }{\mathrm{\nabla }}_{x}u{\mathrm{\parallel }}_2^2$. Facet surface measure scales by the corresponding cofactor, with factor comparable to $h_K^{d-1}$; substituting [F2] into step 1.1 yields the asserted $h_K^{-1}$ and $h_K$ factors.

For smooth-neighborhood approximants $u_n\to u$ in $H^1(K)$ from [F1], step 2.1 applied to $u_n-u_m$ makes their facet restrictions Cauchy in $L^2(F)$. Define ${\gamma }_{F}u$ as this limit. If another such sequence is chosen, the same bound applied to the difference of the sequences shows the same limit, so the trace is well-defined. Passing to the limit in step 2.1 gives the displayed bound for $u$.