Homogenization of non-linear Dirichlet problems in perforated domains of general type

I. V. Skrypnik

Sbornik. Mathematics, Tome 187 (1996) no. 8, pp. 1229-1260

Résumé

A sequence of boundary-value problems for a second-order non-linear elliptic equation in domains $\Omega_s\subset\Omega\subset\mathbb R^n$ and $s=1,2,\dots$ is considered. No geometric assumptions on the $\Omega_s$ are made. The existence of a sequence $r_s$ approaching zero as $s\to\infty$ is assumed such that $C_m\bigl(K(x_0,r)\setminus \Omega_s\bigr)\leqslant Ar^n$ for $r\geqslant r_s>0$ and for an arbitrary point $x_0\in\Omega$. Here $K(x_0,r)$ is the $2r$-cube with centre at $x_0$ and $C_m$ is the $m$-capacity. The conditions imposed on the coefficients of the equation ensure that the energy space is $W_m^1$. The strong convergence of the solutions $u_s(x)$ of the problems under consideration is proved in $W_p^1$ for $p$; a corrector in $W_m^1$ and a homogenized boundary-value problem are constructed. These results are based on an asymptotic expansion for the sequence $u_s(x)$ and on a new pointwise estimate of the solution of a certain model non-linear problem.