Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 107 / 4 / b / ii / Solution

If $u=0$ on $\Sigma$ and $|\Sigma |>0$, thenPart (b.i) consequently givesLetting $\epsilon \downarrow 0$ and using Fatou lemma yields ${\int }_{B_1}{u}^{p_0}=0$. Since $u\ge 0$ is continuous,