Integral convergence of solutions of the Dirichlet problem for a quasilinear elliptic equation in perforated domains

In a family of Lipschitz domains $\Omega_\varepsilon$ of the Euclidean space $\mathbb R^n$ which are obtained by excluding from a bounded domain $\Omega\subset\mathbb R^n$ of a certain cavity having the connected components of characteristic size $\varepsilon$ for each value of the small parameter $\varepsilon$, the Dirichlet problem for a second-order quasilinear elliptic equation with measurable coefficients is considered. The equation contains in its main part an operator of the $p$-Laplacian kind and having a minor term wich depends according to a power law on the sought function. The Dirichlet condition at the boundary of the cavity depends arbitrarily on the small parameter $\varepsilon$, and the condition on the common part of the boundary $\partial \Omega$ of all the domains of the family is homogeneous. Sufficient conditions for convergence to zero as $\varepsilon \to 0$ of the family of solutions $\{u_\varepsilon(x)\}$ of the Dirichlet problem and their gradients are established in an integral norm. An estimate of the convergence rate is obtained. The convergence criterion is refined for the case when the cavities are the union of a finite number of balls of radius $\varepsilon$ or cylinders with a small section. Bibliography: 13 items.