# Squared gradient of a harmonic function is subharmonic

I'm trying to prove the following: let $u\in C^2(\mathbb{R}^n)$ be an harmonic function, then the function $w:=|\nabla(u)|^2$ is subharmonic, that is, $\Delta(w)\geq 0$.

As a matter of fact this exercise has as a previous task to prove that $v(x):=|u|^p$, $p\geq 1$ is subharmonic w.r.t. the mean, that is

$$v(x)\leq \frac{1}{B_R}\int_{B_R}v(y) \ dy$$ I have already proven that with the property of $u$ on the mean value of harmonic functions and Holder inequality. I don't really know if this part is supposed to be related with the other. It would obviously enforce my faith in test makers if it was.

• Can you prove that a partial derivative of a harmonic function also harmonic? This is immediate (by permuting derivative and $\Delta$) if it's already known that harmonic functions are infinitely smooth. – user357151 Aug 17 '17 at 14:55
• Yes, I can do that, by that, the inequality works for w too – ZenoCozeno Aug 17 '17 at 16:23
• see tthat $\nabla u$ is also harmonic and use your statement about the sub-mean value property. – Guy Fsone Aug 17 '17 at 19:16