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

2010/10/01 by Filip Rindler, Rindler, Filip · 1 citation
Mathematics · Medicine · #26B30 #28B05 #49J45 (primary) #Advanced Banach Space Theory #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Muscle and Compartmental Disorders #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1010.0242

openalex publication_date 2010/10/01 · openalex created_date 2025/10/27 · openalex updated_date 2026/07/28

Abstract

We give a new proof of sequential weak* lower semicontinuity in\n BV(\Ω; R^m) for integral functionals with a quasiconvex Carath 'eodory\nintegrand with linear growth at infinity and such that the recession function\nf^\∞ exists in a strong sense and is (jointly) continuous. In contrast to\nthe classical proofs by Ambrosio & Dal Maso [J. Funct. Anal. 109 (1992), 76-97]\nand Fonseca & M "uller [Arch. Ration. Mech. Anal. 123 (1993), 1-49], we do\nnot use Alberti's Rank-One Theorem [Proc. Roy. Soc. Edinburgh Sect. A 123\n(1993), 239-274], but a rigidity result for gradients. The proof is set in the\nframework of generalized Young measures and proceeds via establishing\nJensen-type inequalities for regular and singular points of Du.\n

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