= Solution
The strict maximum principle applied to $a_{ij}\partial_{ij}u=f>0$ with zero boundary data gives $u<0$ in $\Omega$. On $\Omega'$ set $w=v-u$. Then $a_{ij}\partial_{ij}w=0$, while $w=-u\geq0$ on $\partial\Omega'$ and this boundary value is positive somewhere because $\Omega'$ is proper. Hence $w>0$ inside and $w(y_0)=0$. The <Hopf boundary point lemma> at this boundary minimum gives
$$
D_\nu w(y_0)<0.
$$
Therefore $D_\nu v(y_0)-D_\nu u(y_0)<0$, proving the normal derivatives are unequal.
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