If $u=0$ on $\Sigma$ and $|\Sigma|>0$, then
$$
\int_{B_1}u_\varepsilon^{-p_0}\geq|\Sigma|\varepsilon^{-p_0}.
$$
Part (b.i) consequently gives
$$
\int_{B_1}u_\varepsilon^{p_0}leq C|\Sigma|^{-1}\varepsilon^{p_0}.
$$
Letting $\varepsilon\downarrow0$ and using <Fatou lemma> yields $\int_{B_1}u^{p_0}=0$. Since $u\geq0$ is continuous,
$$
\boxed{u\equiv0\text{ on }B_1.}
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-107/4/b/ii/solution