Homogenization of problems with gradient constraints (Q2467892)

Given \(u_{\varepsilon}\) a sequence of solutions of \[ I_{\varepsilon}(u_{\varepsilon}) = \min_{u \in W_0^{1,p}(\Omega)} I_{\varepsilon}(u), \quad I_{\varepsilon}(u) = \int_{ \Omega} \Biggl( f\biggl( {x \over \varepsilon }, \nabla u\biggr) -gu\Biggr)\, dx, \] then, up to a subsequence, \( u_{\epsilon} \rightharpoondown u_0 \) weakly in \(W_0^{1,p} ( \Omega) \) where \(u_0\) is a solution of the homogenized problem of minimization of an integral functional of the calculus of variation whose integrand \( f(y, \xi)\) takes values in \([0,+ \infty]\) and is convex and uniformly \(p\) coercive, \(1 <p < + \infty\), in the \( \xi\) variable. The unboundedness of the integrand can be interpreted as a rapidly oscillating constraint on the gradient of the admissible functions.