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


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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