= 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 $p$,
$$
\widehat u_k(p)=\int_Uu_k(x)e^{-ip\cdot x}\,dx
\longrightarrow
\int_Uu(x)e^{-ip\cdot x}\,dx
=\widehat u(p),
$$
because $e^{-ip\cdot x}\in L^2(U)$.
Fix $t>0$. On the finite-measure ball $|p|\leq t$, the Fourier transforms are uniformly bounded by
$$
|\widehat u_k(p)-\widehat u(p)|
\leq |U|^{1/2}\|u_k-u\|_2.
$$
Dominated convergence therefore gives
$$
\int_{|p|\leq t}|\widehat u_k-\widehat u|^2\,dp\to0.
$$
On the complementary region,
$$
|\widehat u_k-\widehat u|^2
\leq2(|\widehat u_k|^2+|\widehat u|^2),
$$
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> yields
$$
\boxed{\|u_k-u\|_{L^2(U)}\to0.}
$$
This is the same low-frequency compactness mechanism used in the <Fourier proof of Rellich compactness>.
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