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Given $u$ a harmonic function on a ball of radius $r$. i.e.

$$ -\Delta u=0 \qquad{\text{in $B_r(0)$}} $$

Then show that $$ |\nabla u(0)|\leq C \frac{1}{r}\def\avint{\mathop{\,\rlap{-}\!\!\int}\nolimits} \avint_{\partial B_r(0)}|u| $$

It would be a great help if I could set some insight on how this inequality changes in case of a non-homogeneous equation and when there is a $p$-Laplacian.

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  • $\begingroup$ @DavidC.UllrichThanks for the correction, I edited the question. Can you give me some reference for the proof? $\endgroup$ – Harish Dec 5 '18 at 15:54
  • $\begingroup$ It's easy from the Poisson integral formula (applied in, say,, $B(0,r/2)$. $\endgroup$ – David C. Ullrich Dec 5 '18 at 15:56

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