Past exam of the functional analytic methods for partial differential equations course of the University of Oxford / 2023 / 3 / b / i / Solution

Let $w_m=|E_m{|}^{5/6}{u}_m$. By linearity of the weak formulation, $w_m$ is the zero-boundary weak solution with forcing ${\chi }_{E_m}$. The estimate in (a)(iii) therefore givesThe change of variables $x=r_{m}y$ gives $|E_m|=r_m^3|E|>0$, so the division is legitimate. Since $H^1(B)$ is a Hilbert space, weak sequential compactness in a Hilbert space now gives a subsequence and some $u\in H^1(B)$ such thatIn fact $u\in H_0^1(B)$: the Sobolev trace operator is continuous and linear, so its zero kernel is preserved by weak convergence.