Homogenization of Dirichlet problems with nonnegative bounded constraints of the gradient (Q1343306)

The authors consider the sequence of variational problems \[ \min \Biggl\{ \int_ \Omega f(hx, Du) dx+ \int_ \Omega \beta u dx: | Du(x)|\leq \varphi(hx),\;u= 0\text{ on } \partial\Omega\Biggr\}\tag{\({\mathcal P}_ h\)} \] and prove that the solutions \(u_ h\) converge (as \(h\to +\infty\)) to solutions of the problem \[ \min \Biggl\{ \int_ \Omega \overline f(x, Du) dx+ \int_ \Omega \beta u dx: u= 0 \text{ on } \partial\Omega\Biggr\}\tag{\({\mathcal P}\)} \] where, indicating by \(Y\) the unit cube of \(\mathbb{R}^ n\), \(\overline f\) is given by \[ \overline f(z)= \min \Biggl\{ \int_ Y f(y, z+ Dw) dy: | z+ Dw(y)|\leq \varphi(y),\;w \text{ is } Y\text{- periodic}\Biggr\}. \] This problem has been considered by several authors (see references) under more or less strict assumptions on \(\varphi\); here \(\varphi\) is only assumed to be nonnegative, bounded, and periodic. The result is obtained by using \(\Gamma\)-convergence techniques.