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

We give a new proof of sequential weak* lower semicontinuity in $\BV(Ω;\R^m)$ for integral functionals with a quasiconvex Carathéodory integrand with linear growth at infinity and such that the recession function $f^\infty$ exists in a strong sense and is (jointly) continuous. In contrast to the classical proofs by Ambrosio & Dal Maso [J. Funct. Anal. 109 (1992), 76-97] and Fonseca & 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 $Du$.

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