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I remember the following statement is correct but I cannot find a reference for that, can anybody help me to give one?

Let $\Omega\subseteq\mathbb{R}^{n}$ be an open, bounded, smooth domain, $\varphi\in C\left(\partial\Omega\right).$ Let $u$ be the unique solution of the Dirichlet problem $$ \begin{cases} \triangle u & =0\quad\textrm{in }\Omega,\\ u & =\varphi\quad\textrm{on }\partial\Omega. \end{cases} $$ Then there exists $C>0$ s.t $$ \omega_{u}\leq C\omega_{\varphi}, $$ where $\omega_{f}$ denotes the modulus of continuity of $f;$ $\omega_{f}\left(\delta\right):=\sup\left\{ \left|f\left(x\right)-f\left(y\right)\right|:\left|x-y\right|<\delta\right\} .$

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  • $\begingroup$ When bounding $|u(y)-u(x)|$ you can reduce to the case $x\in\partial\Omega$ by the maximum principle, and then $y\in\partial\Omega$ too, via a supersolution (here should appear C). $\endgroup$
    – Pietro Majer
    Commented Jan 29, 2017 at 9:02
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    $\begingroup$ Can u give some more hints. The right hand side is proportional to $\omega_{\varphi}$, not just depending on $\omega_{\varphi}$. $\endgroup$
    – user97743
    Commented Jan 29, 2017 at 10:16
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    $\begingroup$ Sadly, after searching online, I 've learned that this is not correct even $\Omega $ is a ball. $\endgroup$
    – user97743
    Commented Jan 31, 2017 at 22:12
  • $\begingroup$ In fact, following my hints above you just get a mod of cont. $\omega'$ for the solution, but it is not dominated by $\omega$... $\endgroup$
    – Pietro Majer
    Commented Jan 31, 2017 at 22:34
  • $\begingroup$ So a modified question could be: what moduli of continuity for $\phi$ admit solutions with the same m.of (times a constant). $\endgroup$
    – Pietro Majer
    Commented Jan 31, 2017 at 22:41

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