Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 107 / 4 / i / 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 givesThe maximum estimate for the Poisson problem also givesThese estimates prove that $T$ is well defined and give the requested norm control.