Geodesic
Homogenization of Monotone Operators Under Conditions of Coercitivity and Growth of Variable Order
Matematičeskie zametki, Tome 90 (2011) no. 1, pp. 53-69

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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.
Keywords: homogenization of monotone operators, Dirichlet boundary-value problem, coercitivity condition, compensated compactness, Sobolev–Orlicz space.
Mots-clés : elliptic equation 
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     author = {V. V. Zhikov and S. E. Pastukhova},
     title = {Homogenization of {Monotone} {Operators} {Under} {Conditions} of {Coercitivity} and {Growth} of {Variable} {Order}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {53--69},
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     volume = {90},
     number = {1},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2011_90_1_a6/}
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V. V. Zhikov; S. E. Pastukhova. Homogenization of Monotone Operators Under Conditions of Coercitivity and Growth of Variable Order. Matematičeskie zametki, Tome 90 (2011) no. 1, pp. 53-69. http://geodesic.mathdoc.fr/item/MZM_2011_90_1_a6/