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Further results on the homogenization of quasilinear operators. (English) Zbl 0658.35016
Consider the homogenization of a family of quasilinear operators \[ A_{\varepsilon}u=-\operatorname{div} a(x/\varepsilon,u,Du),\quad \varepsilon >0;\quad u\in H_0^{1,p}(\Omega),\quad p>1. \] Using Gehring’s lemma [cf. M. Giaquinta, Mutiple integrals in the calculus of variations and nonlinear elliptic systems. Princeton, New Jersey: Princeton University Press (1983; Zbl 0516.49003)], it is proved that the gradients of the solutions \(u_{\varepsilon}\) of the problem \[ A_{\varepsilon}u=f, \quad f\in L^r,\quad r>p/(p-1);\quad u\in H_0^{1,p}(\Omega),\quad p>1,\quad q>p \] belong to \((L^2(\Omega))^n\).
Reviewer: J. H. Tian

35G20 Nonlinear higher-order PDEs
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35B65 Smoothness and regularity of solutions to PDEs
35B40 Asymptotic behavior of solutions to PDEs