Liftings, Young measures, and lower semicontinuity

TL;DR

This paper introduces liftings and Young measures as new tools to analyze the asymptotic behavior of BV sequences, leading to an improved integral representation theorem for functional relaxation under weaker hypotheses.

Contribution

It presents a novel approach using liftings and Young measures to establish a more general lower semicontinuity result for variational functionals.

Findings

Established an integral representation theorem for relaxed functionals.

Proved lower semicontinuity under minimal hypotheses.

Introduced liftings as a localization tool in variational analysis.

Abstract

This work introduces liftings and their associated Young measures as new tools to study the asymptotic behaviour of sequences of pairs $(u_j,Du_j)j$ for $(u_j)_j \in \mathrm{BV}(\Omega;\mathbb{R}^m)$ under weak* convergence. These tools are then used to prove an integral representation theorem for the relaxation of the functional \[ \mathcal{F}\colon u\to\int_\Omega f(x,u(x),\nabla u(x)) \;\mathrm{dx},\quad u\in\mathrm{W}^{1,1}({\Omega};\mathbb{R}^m),\quad {\Omega}\in\mathbb{R}^d\text{ open}, \] to the space $\mathrm{BV}(\Omega; \mathbb{R}^m)$. 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.