Let
$$
Lu=-a\_{ij}(x)\partial\_{ij}u+b\_i(x)\partial\_i u
$$
be a uniformly elliptic operator, with $A(x)=(a\_{ij}(x))$ positive-definite.
Here I'm only considering smooth coefficients, and the domain $\Omega\subset \mathbb R^d$ is as smooth as needed (but bounded).
In one of its elementary versions, the classical [Hopf boundary point lemma](https://en.wikipedia.org/wiki/Hopf_lemma) states that if $u\in C^2(\Omega)\cap C^1(\bar\Omega)$ is a supersolution
$$
Lu\geq 0
$$
and attains a minimum point at $x\_0\in \partial\Omega$ then
$$
\partial\_\nu u(x\_0)<0,
$$
where $\nu=\nu(x\_0)$ is the outer normal to $\partial\Omega$ at $x\_0$.
In other words, the supersolution must grow linearly inside the domain close to a boundary minimum point.
There are various possible extensions, in particular $\nu$ can be any outward pointing direction, and $\Omega$ can have corners.


**Question:**



> 
> Can one say anything about the behaviour of $u$ at any such boundary minimum point, assuming only that $u\in C^2(\Omega)\cap C(\bar\Omega)$?
> 
> 
> 


The key point here being the lack of $C^1$ regularity up to the boudnary.
I need this typically for a singular eigenvalue problem, where in fact I am trying to retrieve some information on a principal eigenpair $(\lambda\_0,u\_0(x))$ to a singular problem of the form
$$
Lu\_0=\frac{1}{\theta(x)}\lambda\_0 u\_0.
$$
My weight $\theta(x)$ is a given, smooth function that is positive inside $\Omega$ but vanishes linearly, typically $\theta(x)=\operatorname {dist}\_{\partial\Omega}(x)$.


Some probabilistic arguments tell me for free that $u\_0\geq 0$ is nontrivial and $\lambda\_0>0$, with $u\_0$ being moreover $C^2$ in the interior and continuously vanishing at the boundary.
I need to show that the vanishing is linear.
Of course Hopf's lemma immediately pops up to mind, but I really cannot guarantee the $C^1$ regularity up to the boundary (and all my attempts in that direction have failed so far).
Actually for my purposes it would suffice to get
$$
c\operatorname{dist}\_{\partial\Omega}(x)\leq u\_0(x) \leq C\operatorname{dist}\_{\partial\Omega}(x)
$$
in a neighborhood of $x\_0$.
Has anyone ever heard of something like that? For example, sandwiching $u\_0$ between two local lower/upper barriers vanishing linearly would do the trick, but I dont' really know how to do that.


*Final kinky comment:* to tell the truth, really, I'm working in the one-dimensional interval $\Omega=(0,1)$. But the question is so natural that I felt I had to write it in a slightly more general framework. So, if anyone has a specific 1D trick I'll be happy too!



Nonsmooth version of Hopf boundary point lemma
I think this is just the comment following Lemma 3.4 of Gilbarg and Trudinger (specifically equation 3.11).


I should add that lowering the regularity of the boundary seems like a harder problem (and is I think false in the case of a square). GT require an interior sphere condition. L. Rosales (see for instance [Generalizing Hopf’s Boundary Point Lemma](https://doi.org/10.4153/CMB-2017-074-6)) seems to have the best results in this direction.


