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On the homogenization of quasilinear divergence structure operators. (English) Zbl 0636.35027
A family of boundary value problems of the type $(1)\quad -div a(x/\epsilon,u,Du)=f,\quad u\in H_ 0^{1,p}(\Omega)$ is considered, where a(x,u,$$\epsilon)$$ is periodic in x and satisfies suitable growth conditions, $$\epsilon >0$$, $$p>1$$. It is proved that the solutions $$u_{\epsilon}$$ of (1) converge weakly to $$u_ 0$$ satisfying $-div b(u,Du)=f,\quad u\in H_ 0^{1,r}(\Omega),$ and $$a(x/\epsilon,u_{\epsilon},Du_{\epsilon})$$ converge to $$b(u_ 0,Du_ 0)$$ weakly in $$L^{p'},p'=p/(p-1)$$, where b(u,$$\xi)$$ is given by an explicit formula. Moreover, it is shown that certain structure conditions are preserved for the limit operator.
Reviewer: M.Kucera

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs 35A35 Theoretical approximation in context of PDEs 35J25 Boundary value problems for second-order elliptic equations
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