Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 107 / 5 / b / iii / Solution

For $v\in C^0(\overline{B})$, let $T(v)$ be the solution from part (i). The weak maximum principle givesOn this fixed bounded range of values, $\lambda$ and $\Lambda$ have common positive lower and finite upper bounds. The global De Giorgi-Nash-Moser theorem therefore bounds $T(v)$ uniformly in one Hölder space. The compact embedding $C^{0,\mu }(\overline{B})\hookrightarrow C^0(\overline{B})$ makes $T$ compact. Uniform convergence $v_k\to v$, the continuity of $a_{ij}$ on the relevant compact set, energy bounds, and uniqueness of the limiting linear problem show that $T(v_k)\to T(v)$ uniformly, so $T$ is continuous.

The same maximum-principle bound controls every solution of the Leray–Schauder homotopy after using boundary data $t\phi$. Thus the homotopy set is bounded, and the Leray-Schauder fixed point theorem supplies $u=T(u)$. By definition,and $u\in W^{1,2}(B)\cap C^0(\overline{B})$.