Homogenization of integrals with pointwise gradient constraints via the periodic unfolding method (Q997553)

In \textit{D. Cioranescu, A. Damlamian, G. Griso}, [C. R. Math. Acad. Sci. Paris 335, 99--104 (2002; Zbl 1001.49016)] a technique based on periodic unfolding is introduced (an ``unfolding'' operator \(\mathcal{T}_\varepsilon\) is defined in such a way that two-scale convergence of \(w_{\epsilon}\) to \(w\) is equivalent to the weak convergence of \(\mathcal{T}_\varepsilon w_{\varepsilon}\) to \(w\) in an appropriate space). The authors use this technique, recalling its main properties, to study a classical problem, the homogenization of a sequence of functionals of the type \[ F(u) = \int_{\Omega} f \big(x, \nabla u (x) \big) dx \] where the energy density \(f\) is periodic in the first variable, convex and with standard \(p\)-growth conditions in the second and with values in \([0, + \infty]\) (by which a constraint on the gradient follows). They consider both the case in which the sequence of functional is defined in \(W^{1,p}(\Omega)\) and the case in which they are defined in \(W^{1,p}_0(\Omega)\).