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.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 105 / 1 / b / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-105/1/b/i/solution