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Asymptotics of solutions and modeling of the Von Karman equations in a singularly perturbed domain

In a plane domain with several small holes of diameter O(ε), we consider the von Karman 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 |ln 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. Bibliography: 34 titles. Illustrations: 1 figure.

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Correspondence to S. A. Nazarov.

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Translated from Problems in Mathematical Analysis 54, February 2011, pp. 113–143.

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Izotova, O.V., Nazarov, S.A. & Sweers, G. Asymptotics of solutions and modeling of the Von Karman equations in a singularly perturbed domain. J Math Sci 173, 571–608 (2011). https://doi.org/10.1007/s10958-011-0261-6

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Keywords

  • Asymptotic Expansion
  • Dirichlet Problem
  • Biharmonic Equation
  • Asymptotic Modeling
  • Hole Boundary