= Solution
For $w\in X_\delta$, hypothesis (2) gives $F(w)\in C^{0,\alpha}(\overline\Omega)$. Thus $f-F(w)$ is Hölder continuous and the classical <Dirichlet problem> for the <Poisson equation> has a unique solution $u=T(w)\in C^{2,\alpha}(\overline\Omega)$. The <Global Schauder estimate> gives
$$
\|T(w)\|_{C^{2,\alpha}}
\leq C\left(\|F(w)\|_{C^{0,\alpha}}+
\|f\|_{C^{0,\alpha}}+
\|T(w)\|_{C^0}+\|\phi\|_{C^{2,\alpha}}\right).
$$
The maximum estimate for the Poisson problem also gives
$$
\|T(w)\|_{C^0}\leq C\left(\|f\|_{C^0}+\|F(w)\|_{C^0}+\|\phi\|_{C^0}\right).
$$
These estimates prove that $T$ is well defined and give the requested norm control.
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