On the homogenization of quasilinear divergence structure operators. (English) Zbl 0636.35027

Fusco, N.; Moscariello, G.

Ann. Mat. Pura Appl., IV. Ser. 146, 1-13 (1987).

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

Keywords:

second order; quasilinear; homogenization; growth conditions; explicit formula; structure; limit operator

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