Fixed-collar extension of C1alpha conductivities

Since $\overline{{\mathrm{\Omega }}^{\mathrm{\prime }}}$ is compact in $\mathrm{\Omega }$, every boundary point of $\mathrm{\Omega }$ has a neighborhood disjoint from $\overline{{\mathrm{\Omega }}^{\mathrm{\prime }}}$. On the open set $\mathrm{\Omega }\setminus \overline{{\mathrm{\Omega }}^{\mathrm{\prime }}}$, continuity and the almost-everywhere background condition force $\gamma =1$ everywhere: otherwise a sufficiently small ball would have positive measure and $\mathrm{\mid }\gamma -1\mathrm{\mid }$ bounded below by a positive number. Thus ${\gamma }_e$ is identically $1$ in a neighborhood of every boundary point and is $C^1$ globally. Its derivatives vanish off $\mathrm{\Omega }$ and near $\mathrm{\partial }\mathrm{\Omega }$. The same continuity argument promotes the almost-everywhere ellipticity bounds to pointwise bounds in $\mathrm{\Omega }$, and the extension gives the claimed global value and derivative bounds. It also shows ${\gamma }_e-1=0$ outside $\overline{{\mathrm{\Omega }}^{\mathrm{\prime }}}$.

For $x,y\in \mathrm{\Omega }$, the Hölder estimate for each derivative follows from [F1]. For $x,y\mathrm{\notin }\mathrm{\Omega }$, both derivatives are zero. In the remaining case take $x\in \mathrm{\Omega }$ and $y\mathrm{\notin }\mathrm{\Omega }$. The segment from $x$ to $y$ has a first exit point $z\in \mathrm{\partial }\mathrm{\Omega }$, with $\mathrm{\mid }x-z\mathrm{\mid }\le \mathrm{\mid }x-y\mathrm{\mid }$. By step 1.1, ${\mathrm{\partial }}_j{\gamma }_e(z)={\mathrm{\partial }}_j{\gamma }_e(y)=0$. Applying the original Hölder estimate on $\bar{\mathrm{\Omega }}$ to $x,z$ gives $\mathrm{\mid }{\mathrm{\partial }}_j{\gamma }_e(x)-{\mathrm{\partial }}_j{\gamma }_e(y)\mathrm{\mid }\le R\mathrm{\mid }x-z{\mathrm{\mid }}^{\alpha }\le R\mathrm{\mid }x-y{\mathrm{\mid }}^{\alpha }$. These cases exhaust all pairs, establishing the global seminorm and hence $C^{1,\alpha }$ regularity.