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

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- Solution a

Solution

The Weak Harnack inequality states that there are

p > 0

and

C = C (n, λ, Λ)

such that every nonnegative weak supersolution satisfies

(\frac{1}{∣ B _{3/4} ∣} \int_{B_{3/4}} u^{p})^{1/ p} \leq C B_{1/2} in f u .

The radii may be replaced by any fixed nested pair of balls, with the constant adjusted accordingly.

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