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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.

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35J65 Nonlinear boundary value problems for linear elliptic equations