Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 107 / 3

The Alexandrov–Bakelman–Pucci maximum principle and the absence of a zeroth-order term giveThe global Schauder estimate isCombining the two bounds gives $|u{|}_{2,\alpha ;B}\le C|f{|}_{0,\alpha ;B}$.

Use the method of continuity withEvery $L_t$ has the same uniform ellipticity and Hölder coefficient bounds, so the estimates in part (a) hold with one constant independent of $t$. They imply injectivity. The set of $t$ for which $L_t:C_0^{2,\alpha }(B)\to C^{0,\alpha }(B)$ is onto is open by the bounded inverse theorem and a Neumann-series perturbation, and closed by the uniform a priori estimate. It contains $0$ because the Dirichlet Laplacian is bijective. Connectedness of $[0,1]$ therefore gives surjectivity at $t=1$, and $L$ is bijective.

Taylor's theorem and $g(0)=g^{\prime }(0)=0$ giveCombining these with the product estimate in a Hölder space yieldsWritingand using the same product estimate gives

Let $G=L^{-1}$ and use the bound from part (b),On the closed ball ${\mathcal{B}}_{\epsilon }\subset C_0^{2,\alpha }(B)$ defineThe Hölder product estimate and part (i) giveandChoose ${\epsilon }_0$ so that $2M{\epsilon }_0<1/2$, then choose ${\delta }_0$ so that the remaining terms make $T$ preserve ${\mathcal{B}}_{{\epsilon }_0}$ and have contraction constant less than one. The Banach fixed-point theorem gives a unique $u$ in that ball satisfying