\( \newcommand{\R}{\mathbb{R}} \newcommand{\dx}{\mathrm{d}} \)

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 \(L^1(\Omega)\).

**Definition (equiintegrability).** For \(\Omega \subset \mathbb{R}^n\) a family of integrable functions \(\mathcal{U}\subset L^1(\Omega)\) is equiintegrable if the following two conditions hold

- The set \(\mathcal{U}\) is tight, i.e. for any \(\varepsilon > 0\) there exists a measurable set \(A\) with \(|A|<\infty\)

\[ \forall u\in \mathcal{U}: \quad \int_{\Omega\backslash A} | u | < \varepsilon.

\] This condition is trivially true if \(|\Omega| < \infty\). - For any \(\varepsilon>0\) there exists \(\delta >0\) such that for every measurable set \(E\) with \(|E|\leq \delta\)

\[

\forall u\in\mathcal{U}: \quad \int_E | u | \;\dx{x} < \varepsilon.

\]

**Lemma (Equivalent characterisation of equiintegrability).**

Let \(\Omega\subset \R^n\), then \(\mathcal{U}\subset L^1(\Omega)\) is a family of equiintegrable functions if and only if

- the family \(\mathcal{U}\) is tight and
- there exists an increasing superlinear function \(\Psi: [0,\infty)\to [0,\infty]\) such that

\[

\sup_{u\in \mathcal{U}} \int_\Omega \Psi(|u|) \; \dx{x} < \infty .

\]

**Theorem (Dunford-Pettis).** A sequence \((u_n)_{n\in \mathbb{N}} \subset L^1(\Omega)\) converges weakly in \(L^1(\Omega)\) if and only if

- the sequence is \(u_n\) is equibounded in \(L^1(\Omega)\):

\[ \sup_n \Vert u_n \Vert_{L^1(\Omega)} < \infty . \] - and the sequence \(u_n\) is equiintegrable.

**Lemma (weak lower semicontinuity of convex functions).** If \(F:\R \to \R\) is convex and

\[

u_n \rightharpoonup u \quad \text{in } L^1(\Omega).

\]
then

\[ \int F(u) \;\dx{x} \leq \liminf_{n\to \infty} \int F(u_n) \;\dx{x} . \]

### References

- K. H. Karlsen, “Notes on weak convergence (MAT4380 – Spring 2006).” pp. 1–14, 2006
- L. C. Evans.”Weak convergence methods for nonlinear partial differential equations”,

volume 74 of CBMS Regional Conference Series in Mathematics. Published for the

Conference Board of the Mathematical Sciences, Washington, DC, 1990.