In this dissertation, we consider a class of boundary value problems which are posed exterior to a thin domain. We are interested in the asymptotic behavior of the solutions as the thickness of the thin region approaches zero.

A general solution procedure for the class under consideration is proposed. The computational domain is reduced to a finite region by means of integral equations, and a variational formulation for the resulting nonlocal boundary value problem is carefully studied. Certain a priori estimates are established. Thereafter, we scale the domain along the thickness, and use a regular asymptotic expansion to approximate the exact solution. This results in a family of simpler variational problems to be solved. We examine solvability issues for this sequence of problems, and may use the a priori estimates to justify the procedure.

The study is carried out with reference to three specific problems which belong to this class. We first exhaustively study a simple model problem, and apply the techniques developed to subsequently analyse a time-harmonic scattering problem. In both these cases we are able to rigorously justify the asymptotic procedure. We then consider a fluid-structure interaction problem, and extend the general procedure to study it. Finally, some computational results are presented.