Homogenization of Monotone Operators Under Conditions of Coercitivity and Growth of Variable Order

We obtain a homogenization procedure for the Dirichlet boundary-value problem for an elliptic equation of monotone type in the domain $\Omega\subset\mathbb R^d$. The operator of the problem satisfies the conditions of coercitivity and of growth with variable order $p_\varepsilon(x)=p(x/\varepsilon)$; furthermore, $p(y)$ is periodic, measurable, and satisfies the estimate $1<\alpha\le p(y)\le\beta<\infty$, while the parameter $\varepsilon>0$ tends to zero. Here $\alpha$ and $\beta$ are arbitrary constants. Taking Lavrentev's phenomenon into account, we consider solutions of two types: $H$- and $W$-solutions. Each of the solution types calls for a distinct homogenization procedure. Its justification is carried out by using the corresponding version of the lemma on compensated compactness, which is proved in the paper.