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

\begin{equation} v(x)\leq \frac{1}{B_R}\int_{B_R}v(y) \ dy \end{equation} 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.

  • $\begingroup$ 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. $\endgroup$ – user357151 Aug 17 '17 at 14:55
  • $\begingroup$ Yes, I can do that, by that, the inequality works for w too $\endgroup$ – ZenoCozeno Aug 17 '17 at 16:23
  • 1
    $\begingroup$ see tthat $\nabla u $ is also harmonic and use your statement about the sub-mean value property. $\endgroup$ – Guy Fsone Aug 17 '17 at 19:16

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