# estimating the gradient of a harmonic function on a ball

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.

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