I’m reading a note on higher regularity boundary Harnack inequality by D. DE SILVA AND O. SAVIN and I’m kind of confused of the case k=1.

In the paper they used the Hopf lemma to show that $u_\nu>c>0$, but, as the boundary regularity is just $C^{1, \alpha}$, I don’t think that we can directly use Hopf lemma.

I tried to use the transformation of coordinates to do make a better regularity of boundary, but it only works in the divergence form of equations. I have no clue to the non-divergence form. Is there any way to do this estimate?