\( \newcommand{\R}{\mathbb{R}} \newcommand{\dx}{\mathrm{d}} \)
Let \(\Omega\subset \mathbb{R}^n\), a sequence \((u_n)\) converges weakly to \(u\in L^p(\Omega)\) if
\[ \int_\Omega u_n v\;\dx{x} = \int_\Omega u v \;\dx{x} , \qquad \forall v\in L^\infty(\Omega) . \] As usual the convergence is denoted by \(u_n \rightharpoonup u\) in \(L^1(\Omega)\).