 Solution

The strict maximum principle applied to $a_{ij}{\partial }_{ij}u=f>0$ with zero boundary data gives $u<0$ in $\Omega$. On ${\Omega }^{\prime }$ set $w=v-u$. Then $a_{ij}{\partial }_{ij}w=0$, while $w=-u\ge 0$ on $\partial {\Omega }^{\prime }$ and this boundary value is positive somewhere because ${\Omega }^{\prime }$ is proper. Hence $w>0$ inside and $w(y_0)=0$. The Hopf boundary point lemma at this boundary minimum givesTherefore $D_{\nu }v(y_0)-D_{\nu }u(y_0)<0$, proving the normal derivatives are unequal.