I am trying to understand a paper, and have come across an "it is easy to see ..." but i don't find it easy at all to see. As far as I can understand, I see the problem thus:
We have a convex bounded open domain $\Omega \subseteq \mathbb{R}^n$ with boundary $\partial\Omega$ of class $\mathcal{C}^2$and a smooth function $u:\Omega \to \mathbb{R}$. Now $u \to -\infty$ as we approach the boundary. Considering that $\Omega$ is convex, how can we see that $u$ is concave near the boundary?
I have a suspicion that this has something to do with the Hessian $H_u$ of $u$. One theorem that comes to mind is:
Suppose $\Omega$ is convex. A $\mathcal{C}^2$ function $f:\Omega \to \mathbb{R}$ is concave if and only if the Hessian $H_f (x)$ is negative semidefinite for all $x\in \Omega$.
Extra information that I don't see the relevance of yet is that $u$ satisfies
$\Delta u = (V-\lambda) -|\nabla u|^2$ where $V$ is some smooth potential and $\lambda$ is positive.
I also don't think the Hopf boundary lemma helps.
Thanks.