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Advances in Calculus of Variations

Managing Editor: Duzaar, Frank / Kinnunen, Juha

Editorial Board Member: Armstrong, Scott N. / Astala, Kari / Colding, Tobias / Dacorogna, Bernard / Dal Maso, Gianni / DiBenedetto, Emmanuele / Fonseca, Irene / Finster, Felix / Gianazza, Ugo / Gursky, Matthew / Hardt, Robert / Ishii, Hitoshi / Kristensen, Jan / Manfredi, Juan / Martell, Jose Maria / McCann, Robert / Mingione, Giuseppe / Nystrom, Kaj / Pacard, Frank / Preiss, David / Riviére, Tristan / Schaetzle, Reiner / Silvestre, Luis

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1864-8266
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Lower semicontinuity and Young measures in BV without Alberti's Rank-One Theorem

Filip Rindler
Published Online: 2012-03-26 | DOI: https://doi.org/10.1515/acv.2011.008

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

About the article

Received: 2010-02-16

Revised: 2011-02-02

Accepted: 2011-03-03

Published Online: 2012-03-26

Published in Print: 2012-04-01


Citation Information: Advances in Calculus of Variations, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258, DOI: https://doi.org/10.1515/acv.2011.008.

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© 2012 by Walter de Gruyter Berlin Boston. Copyright Clearance Center

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Guido De Philippis and Filip Rindler
Archive for Rational Mechanics and Analysis, 2017, Volume 224, Number 3, Page 1087
[2]
Guido De Philippis and Filip Rindler
Annals of Mathematics, 2016, Volume 184, Number 3, Page 1017
[3]
Jan Kristensen and Panu Lahti
Calculus of Variations and Partial Differential Equations, 2016, Volume 55, Number 3
[4]
José Matias and Marco Morandotti
São Paulo Journal of Mathematical Sciences, 2015, Volume 9, Number 2, Page 162
[5]
Filip Rindler and Giles Shaw
The Quarterly Journal of Mathematics, 2015, Volume 66, Number 3, Page 953
[6]
Jan Kristensen and Filip Rindler
Numerische Mathematik, 2016, Volume 132, Number 2, Page 329
[7]
José Matias, Marco Morandotti, and Pedro M. Santos
Applied Mathematics & Optimization, 2015, Volume 72, Number 3, Page 523
[8]
Nikos Katzourakis
Communications in Partial Differential Equations, 2014, Volume 39, Number 11, Page 2091
[9]
Filip Rindler
Archive for Rational Mechanics and Analysis, 2015, Volume 215, Number 1, Page 1
[10]
Filip Rindler
Journal of Functional Analysis, 2014, Volume 266, Number 11, Page 6335

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