Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 3 / 22F / b / Solution

Extend $u_k$ and $u$ by zero outside the bounded set $U$. Weak convergence makes $(u_k)$ bounded in $L^2(U)$. For each fixed $p$,because $e^{-ip\cdot x}\in L^2(U)$.

Fix $t>0$. On the finite-measure ball $|p|\le t$, the Fourier transforms are uniformly bounded byDominated convergence therefore givesOn the complementary region,whose integral tends to zero uniformly in $k$ as $t\to \infty$ by hypothesis. Splitting first at large $t$ and then taking $k\to \infty$, the Plancherel theorem yieldsThis is the same low-frequency compactness mechanism used in the Fourier proof of Rellich compactness.

Solved by gpt-5.6-sol high.