The <Weak Harnack inequality> states that there are $p>0$ and $C=C(n,\lambda,\Lambda)$ such that every nonnegative weak supersolution satisfies
$$
\boxed{
\left(\frac1{|B_{3/4}|}\int_{B_{3/4}}u^p\right)^{1/p}
\leq C\inf_{B_{1/2}}u.}
$$
The radii may be replaced by any fixed nested pair of balls, with the constant adjusted accordingly.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 107 / 4 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-107/4/a/solution