# Nonsmooth version of Hopf boundary point lemma

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 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!

• @LSpice: thanks for the tex editing, I am indeed too lazy for real operatorname without macros (mathbb R is about as far as I usually go...) As for the text: yes, I really meant kinky, because I'm (poorly) trying to trick people into the question, whereas it's really a 1D ODE in the end. By all means feel free to edit if you feel it appropriate. Sep 11 at 23:04
• I wouldn't want to edit to change meaning. Since the question has been answered anyway, I have deleted my comment. Sep 12 at 0:36
• Leo, perhaps you'll find something useful also in this paper by Alvarado, R.; Brigham, D.; Maz’ya, V.; Mitrea, M.; Ziadé, E. "On the regularity of domains satisfying a uniform hour-glass condition and a sharp version of the Hopf-Oleinik boundary point principle" Journal of Mathematical Sciences (New York) 176, No. 3, 281-360 (2011), MR2839047, Zbl 1290.35046. Sep 12 at 18:37
• it deals with a generalization of the interior sphere condition and offers some sharp results. Sep 12 at 18:41
• Indeed this looks quite interesting to me, thanks @DanieleTampieri Sep 13 at 8:44