Weak semi-continuity of the duality product in Sobolev spaces

Given a weakly convergent sequence of positive functions in , 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 . 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.