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Correctors for the homogenization of monotone operators. (English) Zbl 0733.35005
It is known that the solutions $$u_{\epsilon}$$ of the homogenization of quasi-linear equations $$-div a(x/\epsilon,Du_{\epsilon})=f,$$ with a(x,$$\xi$$) periodic in x and monotonic in $$\xi$$ converge weakly in $$H^{1,p}(\Omega)$$ to a solution u of a limit equation defined only in terms of a(x,$$\xi$$). In this paper the authors find correctors $$v_{\epsilon}$$ defined by $$Dv_{\epsilon}=p(x/\epsilon,(M_{\epsilon}Du)(x))$$ with the following properties: $$v_{\epsilon}$$ converge strongly to u in $$H^{1,p}$$, p depends only upon a(x,$$\xi$$), $$M_{\epsilon}$$ are linear operators depending only upon a(x,$$\xi$$) such that for every $$\phi \in (L^ p(\Omega))^ n$$, $$\Omega \subset R^ n$$, $$M_{\epsilon}\phi$$ is a step function.

##### MSC:
 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 35J65 Nonlinear boundary value problems for linear elliptic equations
##### Keywords:
limit equation; correctors