Variationson a strange semi-continuity result

This paper is devoted to the study of some lower semi-continuity results whose model example is lim inf n→+∞ ∇un · ∇vn ∇u · ∇v, for any sequence (un, vn) which weakly converges to (u, v) in H1 0 ( )2 and satisfies the positivity assumptions − un 0 and − vn 0. Extensions of this semi-continuity result are obtained in the following cases: two fixed linear operators, two monotone operators, two varying linear operators, varying domains and non-uniformly bounded operators. In these various cases, the operators under consideration are second-order elliptic operators in divergence form and truncation arguments play an essential role. Finally, the case of higher-order operators is treated by a quite different approach. © 2004 Elsevier Inc. All rights reserved.