Boundary Integral Equations

In Chapter 1 we presented basic ideas for the reduction of boundary value problems of the Laplacian to various forms of boundary integral equations based on the direct approach. This reduction can be easily extended to more general partial differential equations. Here we will consider, in particular, the Helmholtz equation, the Lamé system, the Stokes equations and the biharmonic equation.

For the Helmholtz equation, we also investigate the solution's asymptotic behavior for small wave numbers and the relation to solutions of the Laplace equation by using the boundary integral equations.

For the Lamé system of elasticity, we first present the boundary integral equations of the first kind as well as of the second kind. Furthermore, we study the behavior of the solution and the boundary integral equations for incompressible materials. As will be seen, this has a close relation to the Stokes system and its boundary integral equations. In the two–sdimensional case, both the Stokes and the Lamé problems can be reduced to solutions of biharmonic boundary value problems which, again, can be solved by using boundary integral equations based on the direct approach.

In this chapter we consider these problems for domains whose boundaries are smooth enough, mostly Lyapounov boundaries, and the boundary charges belonging to Hölder spaces. Later on we shall consider the boundary integral equations again on Sobolev trace spaces which is more appropriate for stability and convergence of corresponding discretization procedures.