Representation Formulae, Local Coordinates and Direct Boundary Integral Equations

In order to generalize the direct approach for the reduction of more general boundary value problems to boundary integral equations than those presented in Chapters 1 and 2 we consider here the 2m–th order positive elliptic systems with real C^∞–coefficients. We begin by collecting all the necessary machinery. This includes the basic definitions and properties of classical function spaces and distributions, the Fourier transform and the definition of Hadamard's finite part integrals which, in fact, represent the natural regularization of homogeneous distributions and of the hypersingular boundary integral operators. For the definition of boundary integral operators one needs the appropriate representation of the boundary manifold Γinvolving local coordinates. Moreover, the calculus of vector fields on Γrequires some basic knowledge in classical differential geometry. For this purpose, a short excursion into differential geometry is included. Once the fundamental solution is available, the representation of solutions to the boundary value problems is based on general Green's formulae which are formulated in terms of distributions and multilayer potentials on Γ. The latter leads us to the direct boundary integral equations of the first and second kind for interior and exterior boundary value problems as well as for transmission problems. As expected, the hypersingular integral operators are given by direct values in terms of Hadamard's finite part integrals.

The results obtained in this chapter will serve as examples of the class of pseudodifferential operators to be considered in Chapters 6–10.