ASYMPTOTICS OF SOLUTIONS AND MODELING OF THE VON KARMAN EQUATIONS IN A SINGULARLY PERTURBED DOMAIN

摘要

In a plane domain with several small holes of diameter 0(ε), we consider the von Kar-man equations describing the flexure of a thin isotropic plate. We construct asymptotic expansions for solutions to the nonlinear and the corresponding linearized problems. The coefficients of expansions turn out to be holomorphic and rational functions of |In ε|~(-1). The asymptotic results for the linear problem (the Kirchhoff plate) are interpreted within the framework of the theory of selfadjoint extensions of differential operators by using the tool of weighted spaces with separated asymptotics. We also present a model of a nonlinear singularly perturbed problem that provides high accuracy asymptotic formulas. This problem includes the generalized Sobolev conditions at the points to which the holes shrink.