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Liftings, young measures, and lower semicontinuity
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Rindler, Filip and Shaw, Giles (2019) Liftings, young measures, and lower semicontinuity. Archive for Rational Mechanics and Analysis, 232 . pp. 1227-1328. doi:10.1007/s00205-018-01343-8 ISSN 0003-9527.
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Official URL: https://doi.org/10.1007/s00205-018-01343-8
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 (uj)j⊂BV(Ω;Rm) under weak* convergence. These tools are then used to prove an integral representation theorem for the relaxation of the functional
F:u↦∫Ωf(x,u(x),∇u(x))dx,u∈W1,1(Ω;Rm),Ω⊂Rd open,
to the space BV(Ω;Rm) . 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 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.
| Item Type: | Journal Article | ||||||||||||
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| Subjects: | Q Science > QA Mathematics | ||||||||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||||||
| Library of Congress Subject Headings (LCSH): | Mathematical analysis, Mechanics | ||||||||||||
| Journal or Publication Title: | Archive for Rational Mechanics and Analysis | ||||||||||||
| Publisher: | Springer | ||||||||||||
| ISSN: | 0003-9527 | ||||||||||||
| Official Date: | 3 June 2019 | ||||||||||||
| Dates: |
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| Volume: | 232 | ||||||||||||
| Page Range: | pp. 1227-1328 | ||||||||||||
| DOI: | 10.1007/s00205-018-01343-8 | ||||||||||||
| Status: | Peer Reviewed | ||||||||||||
| Publication Status: | Published | ||||||||||||
| Access rights to Published version: | Open Access (Creative Commons) | ||||||||||||
| Date of first compliant deposit: | 3 January 2019 | ||||||||||||
| Date of first compliant Open Access: | 28 January 2019 | ||||||||||||
| RIOXX Funder/Project Grant: |
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