Liftings, Young Measures, and Lower Semicontinuity

Abstract

This work introduces liftings and their associated Young measures as new tools to study the asymptotic behaviour of sequences of pairs (uj, Duj)j for \({(u_j)_j\subset {\rm BV}(\Omega;\mathbb{R}^m)}\) under weak* convergence. These tools are then used to prove an integral representation theorem for the relaxation of the functional

$$\mathcal{F}\colon u\mapsto\int_\Omega f(x,u(x),\nabla u(x))\, {\rm d}x, \quad u \in {\rm W}^{1,1}(\Omega;\mathbb{R}^m),\quad\Omega\subset\mathbb{R}^d {\rm open,}$$

to the space \({{\rm BV}(\Omega;\mathbb{R}^m)}\). Lower semicontinuity results of this type were first obtained by Fonseca and Müller (Arch Ration Mech Anal 123:1–49, 1993) 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 \({\mathcal{F}}\) 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.

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Acknowledgements

The authors would like to thank Irene Fonseca, Jan Kristensen and Neshan Wickramasekera for several helpful discussions related to this paper.

Funding

Funding

G.S.’s contribution to this work forms part of their PhD thesis and was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) Grant EP/H023348/1 for the University of Cambridge Centre for Doctoral Training, the Cambridge Centre for Analysis. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, Grant Agreement No 757254 (SINGULARITY). F. R. also acknowledges the support from an EPSRC Research Fellowship on Singularities in Nonlinear PDEs (EP/L018934/1).

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Correspondence to Giles Shaw.

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Communicated by I. Fonseca

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Rindler, F., Shaw, G. Liftings, Young Measures, and Lower Semicontinuity. Arch Rational Mech Anal 232, 1227–1328 (2019). https://doi.org/10.1007/s00205-018-01343-8

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