Asymptotics of solutions and modelling the problems of elasticity theory in domains with rapidly oscillating boundaries

S. A. Nazarov

Izvestiya. Mathematics , Tome 72 (2008) no. 3, pp. 509-564.

Résumé

We obtain explicit formulae for two terms of asymptotics of solutions of the Neumann and Dirichlet problems for the system of two-dimensional equations of elasticity theory in a domain with rapidly oscillating boundary and suggest an algorithm for constructing complete asymptotic expansions. We justify the asymptotic representations of solutions using Korn's inequality in singularly perturbed domains. We discuss two methods of modelling these problems of elasticity theory by constructing new, simpler, boundary-value problems whose solutions provide two-term asymptotics of solutions of the original problems. The first method is based on the introduction of the so-called wall laws containing a small parameter in the higher derivatives. The second method is based on the use of the concept of a smooth image of the singularly perturbed boundary.

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