Convergence of the solutions of a nonlinear Dirichlet problem at the refinement of the boundary of the domain

This paper is devoted to the investigation of the convergence of the solutions of Dirichlet problems for quasilinear second-order elliptic equations in a sequence of domains $\Omega^s$ with a fine-grained boundary in the case of the concentration of the fine-grained boundary near some smooth surface. One indicates conditions under which the solutions of the investigated problems converge for $s\to\infty,$ one investigates the character of the convergence of the solutions, and one obtains a boundary problem for the limit function. It is shown that under certain conditions the solutions of the problems in the domains $\Omega^s$ can be replaced approximately, for large $s$, by the limit function which can be found without solving the sequence of problems in the domains $\Omega^s.$