Let $u\in C^2(D)$, $D$ is the closed unit disk in $\mathbf{R}^2$. Assume that $\Delta u>0$. Show that $u$ cannot have a maximum point in $D\setminus\partial D$.
This statement is in a calculus book, after the discussion of extremal values of multivariable functions. So my guess is that I should use the Hessian of $u$ somehow. I started to proof indirectly. Assume that $(x_0,y_0)\in D\setminus\partial D$ is a maximum point. Then $\frac{\partial u}{\partial x}(x_0,y_0),\,\frac{\partial u}{\partial y}(x_0,y_0)=0$. Now I want to investigate the positive/negative definiteness of Hessian and deduce contradiction, but I got stuck.