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Advances in Calculus of Variations

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Lower semicontinuity and Young measures in BV without Alberti's Rank-One Theorem

1Mathematical Institute, University of Oxford, 24–29 St Giles', Oxford OX1 3LB, United Kingdom

Citation Information: Advances in Calculus of Variations. Volume 5, Issue 2, Pages 127–159, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258, DOI: 10.1515/acv.2011.008, March 2012

Publication History

Received:
2010-02-16
Revised:
2011-02-02
Accepted:
2011-03-03
Published Online:
2012-03-26

Abstract.

We give a new proof of sequential weak* lower semicontinuity in for integral functionals of the form

where and is a quasiconvex Carathéodory integrand with linear growth at infinity, i.e. for some , and such that the recession function exists and is (jointly) continuous. In contrast to the classical proofs by Ambrosio and Dal Maso [J. Funct. Anal. 109 (1992), 76–97] and Fonseca and Müller [Arch. Ration. Mech. Anal. 123 (1993), 1–49], we do not use Alberti's Rank-One Theorem [Proc. Roy. Soc. Edinburgh Sect. A 123 (1993), 239–274], but a rigidity result for gradients. The proof is set in the framework of generalized Young measures and proceeds via establishing Jensen-type inequalities for regular and singular points of .

Keywords.: BV; lower semicontinuity; Alberti's Rank-One Theorem; rigidity; Young measure; differential inclusion

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