Let $\Omega=B_1(0)$ and $u\in C(\Omega, R^n) \cap C^2(\Omega, R^n)$ be a vector valued map into the unit ball ( ie. $|u(x)|\le 1$ for all $x\in \Omega $, such that $|\triangle u(x)| \le |\triangledown u(x)|^2$ for all $x\in \Omega$
How can i show that $v:= |u^2| $ is a subharmonic and conclude that $sup_\Omega|u| \le sup_{\partial \Omega } |u| $ .
Thank you for your help .