Integral Equations on Γ⊂ IR³ Recast as Pseudodifferential Equations

The treatment of boundary value problems in Chapter 5 was based on variational principles and lead us to corresponding boundary integral equations in weak formulations. The mapping properties of the boundary integral operators were derived from the variational solutions of the corresponding partial differential equations. This approach is restricted to only those boundary integral operators associated with boundary value problems which can be formulated in terms of general variational principles based on Gårding's inequality.

On the other hand, the boundary integral operators can also be considered as special classes of pseudodifferential operators. In the previous chapters 6 to 8, we have presented some basic properties of pseudodifferential operators. The purpose of this chapter is to apply the basic tools from previous pseudodifferential operator theory to concrete examples of this class of boundary integral equations for elliptic boundary value problems in applications. In particular, the three–dimensional boundary value problems for the Helmholtz equation in scattering theory, the Lamé equations of linear elasticity and the Stokes system will serve as model problems. Two–dimensional problems will be pursued in the next chapter.