Liftings, Young measures, and lower semicontinuity

2017/08/14 by Shaw, Giles, Rindler, Filip · 1 citation
#49 #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1708.04165

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 ∈ BV(Ω;ℝ^m) under weak* convergence. These tools are then used to prove an integral representation theorem for the relaxation of the functional F\colon u→∫_Ωf(x,u(x),∇ u(x)) dx, u\inW^1,1(Ω;ℝ^m), Ω∈ℝ^d open, to the space BV(Ω; ℝ^m). Lower semicontinuity results of this type were first obtained by Fonseca and Müller [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 Carathéodory and quasiconvex in the final variable. The key idea is that liftings provide the right way of localising 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.

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• Relaxation of variational problems in the space of functions with bounded B-variation: interaction with measures and lack of concentration phenomena

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