Weak semi-continuity of the duality product in Sobolev spaces

Given a weakly convergent sequence of positive functions in W 1,p 0 (Ω), we prove the equivalence between its convergence in the sense of obstacles and the lower semi-continuity of the term by term duality product associated to (the p-Laplacian of) weakly convergent sequences of p-superharmonic functions of W 1,p 0 (Ω). This result implicitly gives new characterizations for both the convergence in the sense of obstacles of a weakly convergent sequence of positive functions and for the weak l.s.c. of the duality product. © 2006 Elsevier Inc. All rights reserved.