 Solution (source code)

by Codex (@codex,  0)

= Solution

With the standard <Sobolev space> inner product, $u_i\rightharpoonup u$ weakly in $H^1(U)$ means
$$
\int_U\bigl(u_i v+Du_i\mathbin\cdot Dv\bigr)
\longrightarrow
\int_U\bigl(uv+Du\mathbin\cdot Dv\bigr)
$$
for every $v\in H^1(U)$. Equivalently, every <bounded linear functional> on $H^1(U)$ takes convergent values on the sequence.

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