manuscripta math. 112, 313–323 (2003) © Springer-Verlag 2003 Nicola Fusco · Flavia Giannetti · Anna Verde A remark on the L 1 -lower semicontinuity for integral functionals in BV Received: 16 May 2002 / Published online: 15 October 2003 Abstract. We study the L 1 -lower semicontinuity in BV of an integral functional of the type   f (x, u, ∇u)dx +   -  u + (x) u - (x) f ∞ (x,t, D s u |D s u| )dtd |D s u|. Our assumptions on f extend previous results recently obtained by Gori, Maggi and Marcellini in the case where the above functional is restricted to W 1,1 . 1. Introduction The L 1 -lower semicontinuity of an integral functional G of the type G(u, ) =   f (x, u, ∇u)dx, (1) where  is an open subset of R N and u ∈ W 1,1 (), has been extensively studied in the past years. The starting point of most of the recent studies on this subject is a celebrated result by Serrin. In [11] he proved that the functional G is lower semicontinuous in W 1,1 () with respect to the L 1 -convergence of u under the assumptions that f :  × R × R N → [0, +∞) is continuous, (2) f(x,t, ·) is convex in R N for every (x,t) ∈  × R, (3) and that one of the following conditions holds: (i) f(x,t,ξ) → +∞ as |ξ | → +∞ for every (x,t) ∈  × R; (ii) f(x,t, ·) is strictly convex for every (x,t) ∈  × R; (iii) the derivatives f x , f ξ , f xξ exist and are continuous. After Serrin’s paper, many authors have generalized his result by weakening either the continuity assumption on f or one of the conditions (i)–(iii) above (see for instance [5], [6], [1], [3], [7], [8], [9], [10], [4]). In particular, two recent papers, one by Gori and Marcellini ([9]), the other one by Gori, Maggi and Marcellini ([10]), have shown that condition (iii) can be replaced by a significantly weaker N. Fusco, F. Giannetti, A. Verde: Dipartimento di Matematica e Applicazioni, Via Cintia, 80126 Napoli, Italy. e-mail: {n.fusco,giannett,anverde@unina.it} DOI: 10.1007/s00229-003-0400-6