Let U $\subset R^n$ be open, and $u:U \Rightarrow$ R be harmonic and nonnegative. Prove that
$|Du(x_0)| \le \frac{n}{r} u(x_0)$, $\forall x_0 \in U$, $\forall B(x_0,r) \subset U$
I really need someone's help.
Thanks a lot
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Think this is discussed quite well in Gilbarg and Trudinger chapter 2 – Baron Mingus Oct 10 '13 at 16:30
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Actually there is a very similar proof in Evans's book but they are different. Could you give me some ideas? Thanks – Yang Oct 10 '13 at 16:36
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Where did you found this statament? – Tomás Oct 10 '13 at 22:09
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In Interior gradient bound you will find a detailed proof of the estimate $$|Du(x_0)|\le \frac{n}{r}[\sup_{U} u - u(x_0)] \tag{1}$$ This is essentially equivalent to the estimate you want. Indeed, given a nonnegative harmonic function $u$, apply (1) to $-u$: $$|Du(x_0)|\le \frac{n}{r}[\sup_{U} (-u) + u(x_0)] \le \frac{n}{r}u(x_0) $$ as desired.
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