Liftings, Young measures, and lower semicontinuity

This work introduces liftings and their associated Young measures as new tools to study the asymptotic behaviour of sequences of pairs $(u_j,D{u}_j)j$ for $(u_j{)}_j\in \mathrm{B}\mathrm{V}(\mathrm{\Omega };{\mathbb{R}}^m)$ under weak* convergence. These tools are then used to prove an integral representation theorem for the relaxation of the functional $$\mathcal{F}\text{\hspace{0.17em}⁣}:u\to {\int }_{\mathrm{\Omega }}f(x,u(x),\mathrm{\nabla }u(x))\mathrm{d}\mathrm{x},u\in {\mathrm{W}}^{1,1}(\mathrm{\Omega };{\mathbb{R}}^m),\mathrm{\Omega }\in {\mathbb{R}}^d\text{ open},$$ to the space $$\mathrm{B}\mathrm{V}$(\Omega; $\mathbb{R}$^m)$. Lower semicontinuity results of this type were first obtained by Fonseca and M"uller [Arch. Ration. Mech. Anal. 123 (1993), 1-49] and later improved by a number of authors, but our theorem is valid under more natural, essentially optimal, hypotheses than those currently present in the literature, requiring principally that $f$ be Carath'eodory and quasiconvex in the final variable. The key idea is that liftings provide the right way of localising $\mathcal{F}$ in the $x$ and $u$ variables simultaneously under weak* convergence. As a consequence, we are able to implement an optimal measure-theoretic blow-up procedure.