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?



About the proof of higher regularity boundary Harnack inequality
*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 $C^{1,\alpha}$.




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*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](https://doi.org/10.2140/apde.2016.9.439), [arXiv:1503.02179](https://arxiv.org/abs/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.


