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