Use the <Weak Harnack inequality>: for some $q>1$ and every nonnegative weak supersolution,
$$
\|u\|_{L^q(B_{1/2})}\leq C\inf_{B_{1/4}}u,
$$
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 gives
$$
\sup_{B_{1/4}}u
\leq C\|u\|_{L^q(B_{1/2})}
\leq C\inf_{B_{1/4}}u.
$$
This is the <Harnack inequality for uniformly elliptic divergence-form equations>.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-107/3/a/ii/solution