Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 107 / 3 / a

First replace $u$ by $u+\epsilon$ and later let $\epsilon \downarrow 0$. In the weak subsolution inequality use the admissible truncations approximating ${\eta }^2{u}^{\alpha -1}$. Uniform ellipticity, the coefficient bound, Cauchy-Schwarz inequality, and Young inequality giveand thereforeApply the Sobolev embedding theorem to $w=\eta u^{\alpha /2}$. The preceding estimate yields, for concentric balls $B_r\subset B_R\subset B_1$,Starting with $\alpha =p>1$, taking ${\alpha }_k=p{\sigma }^k$, and choosing radii decreasing to $1/2$, the product of constants converges because ${\sum }_k{\alpha }_k^{-1}<\infty$. Letting $k\to \infty$ provesThis exponent-raising argument is Moser iteration.

Use the Weak Harnack inequality: for some $q>1$ and every nonnegative weak supersolution,where $C$ and $q$ depend only on $n,\lambda ,\Lambda$. A weak solution is both a subsolution and a supersolution. Applying part (i), after rescaling from $B_1$ to $B_{1/2}$, and then the weak Harnack inequality givesThis is the Harnack inequality for uniformly elliptic divergence-form equations.