Variations on a strange semi-continuity result

This paper is devoted to the study of some lower semi-continuity results whose model example isliminfn→+∞∫Ω∇un·∇vn⩾∫Ω∇u·∇v,for any sequence (un,vn) which weakly converges to (u,v) in H01(Ω)2 and satisfies the positivity assumptions -Δun⩾0 and -Δvn⩾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.