A compactly supported smooth conductivity in the higher-dimensional class

By [F2] with radii $1\mathrm{/}2<1$, a bump $\psi$ as in the Example exists; it is nonzero since it equals one on $B(0,1\mathrm{/}2)$. Its first and second derivatives are bounded because it is smooth with compact support. If $\mathrm{\mid }x-y\mathrm{\mid }\le 1$, the mean-value estimate gives $\mathrm{\mid }\mathrm{\nabla }\psi (x)-\mathrm{\nabla }\psi (y)\mathrm{\mid }\le \mathrm{\parallel }{D}^2\psi {\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\mid }x-y\mathrm{\mid }\le \mathrm{\parallel }{D}^2\psi {\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\mid }x-y{\mathrm{\mid }}^{\alpha }$; if $\mathrm{\mid }x-y\mathrm{\mid }\ge 1$, the quotient is at most $2\mathrm{\parallel }\mathrm{\nabla }\psi {\mathrm{\parallel }}_{\mathrm{\infty }}$. Thus $0<N_{\alpha }<\mathrm{\infty }$ and the normalization yields $0\mathrm{\ne }\varphi$ with $\mathrm{\parallel }\varphi {\mathrm{\parallel }}_{C^{1,\alpha }}\le 1$.

The support of $x\mapsto \varphi ((x-x_0)\mathrm{/}r)$ lies in $B(x_0,r)\subset D^{\mathrm{\prime }}$, so $\gamma =1$ on $D\setminus D^{\mathrm{\prime }}$. Since $\mathrm{\mid }\varphi \mathrm{\mid }\le 1$ and $\epsilon \le 1\mathrm{/}4$, its values lie in $[3\mathrm{/}4,5\mathrm{/}4]\subset [1\mathrm{/}2,3\mathrm{/}2]$.

The sup norm of the perturbation is at most $\epsilon \le 1\mathrm{/}4$. Its first derivative suprema are at most $\epsilon \mathrm{/}r\le r^{\alpha }\mathrm{/}8\le 1\mathrm{/}8$, and the $\alpha$-Hölder seminorm of each first derivative is at most $\epsilon r^{-1-\alpha }[\mathrm{\nabla }\varphi {]}_{C^{\alpha }}\le 1\mathrm{/}8$ by scaling the Hölder quotient. The derivatives of the constant background vanish, so $\mathrm{\parallel }\gamma {\mathrm{\parallel }}_{C^{1,\alpha }(\bar{D})}\le 5\mathrm{/}4<2$. All conditions in [F1] hold, hence $\gamma \in X_d(2)$.