Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 107 / 4

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.

For $w\in X_{\delta }$, hypothesis (2) givesCombining the two estimates in part 4(i) with (5) yieldsChoose $0<\delta <\min \{1,(2C{)}^{-1}\}$, so that $C{\delta }^2\le \delta /2$, and then choose $\epsilon \le \delta /(2C)$. Uniformly for $w\in X_{\delta }$ we obtain $\Vert T(w){\Vert }_{C^{2,\alpha }}\le \delta$. Hence

As printed, the hypotheses admit no function $F$. Take the constant function $g\equiv z$ in (2). Its Hölder seminorm vanishes, soBut (1) simultaneously requires $F(z)\ge |z|$. For every $0<|z|<1$, this would give $|z|\le z^2$, which is false. Therefore no $F\in C^{\infty }(\mathbb{R})$ satisfies (1) and (2), and the contraction question is vacuous under the stated assumptions. This contradiction is present in the official paper, rather than arising from the text conversion.