DOI: 10.1007/s11587-006-0003-0 Ricerche di Matematica 55: 31–53 (2006) Doina Cioranescu · Alain Damlamian · Riccardo De Arcangelis Homogenization of integrals with pointwise gradient constraints via the periodic unfolding method Received: February 23, 2006 / Accepted: March 23, 2006 Abstract The pointwise gradient constrained homogenization process, for Neu- mann and Dirichlet type problems, is analyzed by means of the periodic unfolding method recently introduced in [21]. Classically, the proof of the homogenization formula in presence of pointwise gradient constraints relies on elaborated mea- sure theoretic arguments. The one proposed here is elementary: it is based on weak convergence arguments in L p spaces, coupled with suitable regularization techniques. Keywords Homogenization · Gradient constrained problems · Periodic unfolding method Mathematics Subject Classification (2000) 49J45 · 35B27 · 74Q05 1 Introduction The homogenization of periodic structures has been carried out in the last thirty years for various kinds of problems involving differential equations and systems, Communicated by the Editor-in-Chief D. Cioranescu Universit´ e Pierre et Marie Curie (Paris VI), Laboratoire Jacques-Louis Lions, 175, rue du Chevaleret, 75013 Paris Cedex 05, France E-mail: cioran@ann.jussieu.fr A. Damlamian Universit´ e Paris XII - Val de Marne, Centre de Math´ ematiques, 61, avenue du G´ en´ eral de Gaulle, 94010 Cr´ eteil Cedex, France E-mail: damlamian@univ-paris12.fr R. De Arcangelis Universit` a di Napoli Federico II, Dipartimento di Matematica e Applicazioni Renato Cacciop- poli, via Cintia, Complesso Monte S. Angelo, 80126 Napoli, Italy E-mail: dearcang@unina.it