Dunford-Pettis theorem

On a finite-measure space, a bounded subset of $L^1$ is relatively weakly compact exactly when it is uniformly integrable. In particular every uniformly integrable bounded sequence has a weakly convergent subsequence. This is the compactness substitute for reflexivity at exponent one. A superlinear integral bound ${\sup }_j\int H(u_j)<\infty$, with $H\ge 0$ and $H(t)/|t|\to \infty$, gives the required uniform integrability: for $H(t)\ge A|t|$ outside $[-R_A,R_A]$,Choose $A$ large and then the measure of $E$ small. This is the direction used in the direct method in the calculus of variations; a mere bounded $L^1$ norm does not suffice, as concentrating spikes show.

For a uniformly integrable bounded sequence of derivatives on a compact interval and a fixed initial value, its primitives are uniformly bounded and equicontinuous. The increment over any short interval is bounded by the integral of the absolute derivative on that interval. The Arzelà-Ascoli theorem gives relative uniform compactness. If the derivatives converge weakly in $L^1$, tests by interval indicators identify every pointwise limit with $u(x)=\alpha +{\int }_a^{x}v$. Hence every convergent subsequence has this same limit and the primitives converge uniformly. The limit belongs to $W^{1,1}$ and retains any common terminal value. Uniform integrability alone gives no essential-supremum bound on its derivative, so this compactness statement does not assert a Lipschitz limit.