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

TL;DR

This paper presents a new proof of lower semicontinuity in BV spaces for certain integral functionals, avoiding Alberti's Rank-One Theorem by using a rigidity result and Young measures.

Contribution

It introduces a novel proof technique for lower semicontinuity in BV without relying on Alberti's Rank-One Theorem, utilizing rigidity results and generalized Young measures.

Findings

New proof of lower semicontinuity in BV spaces

Avoids use of Alberti's Rank-One Theorem

Uses rigidity results and Young measures

Abstract

We give a new proof of sequential weak* lower semicontinuity in $\BV(\Omega;\R^m)$ for integral functionals with a quasiconvex Carath\'{e}odory 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 & M\"{u}ller [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 $Du$.