Let $\partial\Omega $ be a compact hypersurface in $\mathbb{R}^{n} $, $ f: \partial \Omega \to \mathbb {R} $ a non-negative function. It is possible to find another compact hypersurface $\Gamma = \partial \Omega '\subset \Omega $, such that the problem below is soluble
\begin{cases}
\text{$\Delta u = 0$ in $\Omega\setminus \Omega'$} \\
\text{$u=f$ in $\partial \Omega$} \\
\text{$u=0$ , $u_{\eta}=1$ in $\Gamma$ ,} \\
\end{cases}
where $\eta$ is the inner normal vector to $\Gamma$?
If we extend $u$ by zero inside $\Omega'$, thus $u\geq 0$ in $\Omega$. The free boundary of the problem, the surface $\Gamma$ becomes $\partial\{u>0\}$.
Aff: Notice that, because of Hopf's maximum principle, the best regularity one should hope for u in , is Lipschitz continuity.
My question is how to use th Hopf lemma for the show that statement????