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 $\Omega^s.$