Today I learned that, if $w$ (a real function defined on a region of $\mathbb R^n$) is a harmonic function, then $U:=\left|\nabla u\right|^2$ is subharmonic. This result makes me wonder whether there is a condition that is sufficient for $U$ to be written in that form.
In other words, given a real function $U$ defined on some region of $\mathbb R^n$, how can we determine whether there exists (at least locally) a harmonic function $u$ such that $U=\left|\nabla u\right|^2$?