Homogenization of monotone operators under conditions of coercitivity and growth of variable order
Vasilii Vasil'evich Zhikov, Svetlana Evgenievna Pastukhova · Mathematical Notes · 2011
We obtain a homogenization procedure for the Dirichlet boundary-value problem for an elliptic equation of monotone type in the domain Ω ⊂ ℝ d . The operator of the problem satisfies the conditions of coercitivity and of growth with variable order p ɛ (x) = p(x/ɛ); furthermore, p(y) is periodic, measurable, and satisfies the estimate 1 0 tends to zero. Here α and β are arbitrary constants. Taking Lavrent’ev’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.