Abstract :
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.
