Homogenization and numerical homogenization of nonlinear equations

In this section, we briefly summarize the properties of monotone and pseudomonotone operatorsMonotone operator. We consider two operators $$a(x,\eta ,\xi )$$ and $$a_{0}(x,\eta ,\xi )$$ , $$\eta \in R$$ and $$\xi \in R^d$$ . We assume that these operators satisfy the following conditions. where $$s>0$$ , $$p>1$$ , $$s\in (0,\min (p-1,1))$$ and $$\nu $$ is the modulus of continuity, a bounded, concave, and continuous function in $$R_+$$ , such that $$\nu (0)=0$$ , $$\nu (t)=1$$ for $$t \ge 1$$ and $$\nu (t)>0$$ for $$t>0$$ .