$\newcommand{\avint}{⨍}$ Let $B_r$ be a call of radius $r$ and centre origin and $k<1$.$w$ satisfy the following PDE: $$ \begin{cases} -\Delta w = 0 \qquad \mbox{in $B_r\setminus B_{kr}$}\\ w=0 \qquad \mbox{on $\partial B_{kr}$}\\ w=\varphi\qquad \mbox{on $\partial B_r$} \end{cases} $$

Show that, for all $x\in \partial B_{kr}$, $$ |\nabla w(x)| \,\le\, \frac{C}{r} \avint_{\partial B_r} \varphi\,d\sigma $$

$d\sigma$ is the surface measure.

(one can take $\varphi\geq 0$ is it helps in the proof. The issue has occured in the Theorem 3.1 of [1], page 436, the estimate mentioned between equation 3.2 and 3.3)

[1] *Alt, Hans Wilhelm; Caffarelli, Luis A.; Friedman, Avner*, **Variational problems with two phases and their free boundary**, Trans. Am. Math. Soc. 282, No. 2, 431-461 (1984). ZBL0844.35137.

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