About the proof of higher regularity boundary Harnack inequality 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?
 A: Edit: The result is fine: Hopf's lemma was proved in

*

*G. Giraud, Problèmes de valeurs à la frontière relatifs à certaines donn ás discontinues, Bull. de la Soc. Math. de France, 61 (1933), 1–54

Below is my incorrect answer (which I keep for reference), where I mistakenly assumed that $C^{1,Dini}$ is stronger than $C^{1,\alpha}$.

Old answer:
Looks like you are right: the regularity assumption on the boundary is insufficient, although I did not check this very carefully.
You may have a look at the paper A counterexample to the Hopf-Oleinik lemma (elliptic case) by D. E. Apushkinskaya and A. I. Nazarov, DOI:10.2140/apde.2016.9.439, arXiv:1503.02179. Let me quote from p. 2 of this paper:

The reduction of the assumptions on the boundary of $\Omega$ up to $C^{1,Dini}$-regularity was realized for various elliptic operators in the papers [Wid67], [Him70] and [Lie85] (see also [Saf08]). A weakened form of the Hopf-Oleinik lemma (the existence of a boundary point $x_1$ in any neighborhood of $x_0$ and a direction $\ell$ such that $\frac{\partial u}{\partial \ell}(x_1) \ne 0$) was proved in [Nad83] for a much wider
class of domains including all Lipschitz ones. We mention also the paper
[Swe97] where the behavior of superharmonic functions near the boundary
of 2-dimensional domains with corners is described in terms of the main
eigenfunction of the Dirichlet Laplacian.
The sharpness of some requirements was confirmed by corresponding
counterexamples constructed in [Wid67], [Him70], [KH75], [Saf08], [ABM$^+$11] and [Naz12]. In particular, the counterexamples from [Wid67], [Him70] and
[Saf08] show that the Hopf-Oleinik result fails for domains lying entirely in
non-Dini paraboloids.

Later, the paper shows that lack of "Dini condition" on the boundary invalidates the assertion of Hopf's lemma.
