Innovative solution of a 2D elastic transmission problem¶

¶Dedicated to Wolfgang L. Wendland on the occasion of his 70th birthday.

Abstract

This article is concerned with a boundary-field equation approach to a class of boundary value problems exterior to a thin domain. A prototype of this kind of problems is the interaction problem with a thin elastic structure. We are interested in the asymptotic behavior of the solution when the thickness of the elastic structure approaches to zero. In particular, formal asymptotic expansions will be developed, and their rigorous justification will be considered. As will be seen, the construction of these formal expansions hinges on the solutions of a sequence of exterior Dirichlet problems, which can be treated by employing boundary element methods. On the other hand, the justification of the corresponding formal procedure requires an independence on the thickness of the thin domain for the constant in the Korn inequality. It is shown that in spite of the reduction of the dimensionality of the domain under consideration, this class of problems are, in general, not singular perturbation problems, because of appropriate interface conditions.

¶Dedicated to Wolfgang L. Wendland on the occasion of his 70th birthday.

Keywords:

• Nonlocal boundary value problem
• Variational formulation
• Asymptotic expansions
• Boundary integral equations

Keywords:

• 1991 Mathematics Subject Classifications: Primary 35J20
• Secondary 41A60
• 45A05

Acknowledgments

This research was supported in part by the German Research Foundation DFG under the Grant SFB 404 Multifield Problems in Continuum Mechanics.

Notes

¶Dedicated to Wolfgang L. Wendland on the occasion of his 70th birthday.