with its boundary $\Gamma$ composed by $\Gamma_D, \Gamma_N, \Gamma_{C'}, \Gamma_{C^{\prime\prime}}$. On $\Gamma_D$, we have $u=0$; On $\Gamma_N$, we have $g=\bm{n}\cdot A \nabla u$; On $\Gamma_{C'}$, a complete Robin condition is imposed, means $-\bm{n}\cdot A \nabla u = u$; while on $\Gamma_{C^{\prime\prime}}$, it is instead with a partial Robin condition $-\bm{n}\cdot A \nabla u = 0$ if $u<0$ and $-\bm{n}\cdot A \nabla u = u$ if $u\geq 0$, or write shortly as $-\bm{n}\cdot A \nabla u = u^+$.

One can check in this problem $j^0(\gamma_ju; \gamma_j v)=\int_{\Gamma_{C'}}uv+\int_{\Gamma_{C^{\prime\prime}}}u^+v $. The original locally Lipschitz function $j(u)=1/2 \int_{\Gamma_{C'}}u^2+1/2\int_{\Gamma_{C^{\prime\prime}}} {\left( u^+ \right)}^2$, and $\partial j (\gamma_ju)=u$ on $\Gamma_{C'}$ and $\partial_j(\gamma_ju)=u^+$ on $\Gamma_{C^{\prime\prime}}$. $\partial j, j^0$ coincide respectively with classical Gâteaux and Frèchet derivative of functional $j(u)$.

(0, 0) rectangle (4, 4); in [1,...,13]{} [ (0.3\*-0.05, 4) – +(135:0.3); (0, 0.3\*-0.05) – +(-0.3, 0); (4, 0.3\*-0.05) – +(0.3, 0); ]{} (-0.5, -0.5) rectangle (2, 0); (2, -0.5) rectangle (4.5, 0); at (0.4, 2) [$\Gamma_N$]{}; at (3.6, 2) [$\Gamma_N$]{}; at (2, 3.6) [$\Gamma_D$]{}; at (1, 0.4) [$\Gamma_{C'}$]{}; at (3, 0.4) [$\Gamma_{C^{\prime\prime}}$]{}; at (2, 2) [$\Omega$]{};

in [1,...,8]{} in [1,...,8]{} [ (0.5\*-0.5, 0.5\*-0.5) rectangle (0.5\*, 0.5\*); (0.5\*-0.25, 0.5\*-0.25) circle (0.15); ]{}

Homogenization {#sec:homogenization}
==============

In this section, we 