= 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 gives
$$
\|w_m\|_{H^1(B)}\leq C|E_m|^{5/6},\qquad
\boxed{\|u_m\|_{H^1(B)}\leq C.}
$$
The <change of variables> $x=r_my$ 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 that
$$
\boxed{u_{m_j}\rightharpoonup u\quad\text{in }H^1(B).}
$$
In fact $u\in H_0^1(B)$: the <Sobolev trace operator> is continuous and linear, so its zero kernel is preserved by <weak convergence>.
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