I read a proof in my book on pde which I find a bit strange. Let $\Omega$ be some bounded domain. For $f\in\mathscr C^2(\Omega)\cap\mathscr C^0(\Omega)$ satisfying $\Delta f\geq0$ it holds that $\max_\bar\Omega(f)=\max_{\partial\Omega}(f)$. The proof follows by showing that it holds for $f$ with $\Delta f>0$ based on the second derivative test, and then generalizes it by considering the function $v(x)=f(x)+\epsilon|x|^2$ for $\epsilon$>0.
My question is how to use the second derivative test?I think we should use Hessian Matrix, but the $\Delta f$ is the addition of second derivatives. How to use second derivative test here?
If $p \in \Omega$ suppose $p$ is a maximum point of $f$.
Then $\frac{\partial f}{\partial x_i}(p) =0$ and $\frac{\partial^2 f}{\partial x_i^2}(p) \leq 0 $ for every $1\leq i\leq n$ thus $\Delta f (p) = \sum_{i=1}^n \frac{\partial^2 f}{\partial x_i^2}(p)\leq 0$