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1. Introduction

This chapter serves as a basic introduction to the reduction of elliptic bound-
ary value problems to boundary integral equations. We begin with model
problems for the Laplace equation. Our approach is the direct formulation
based on Green’s formula, in contrast to the indirect approach based on a
layer ansatz. For ease of reading, we begin with the interior and exterior
Dirichlet and Neumann problems of the Laplacian and their reduction to
various forms of boundary integral equations, without detailed analysis. (For
the classical results see e.g. G¨unter [113] and Kellogg [155].) The Laplace
equation, and more generally, the Poisson equation,

−∆v = f in Ω or Ω c

already models many problems in engineering, physics and other disciplines
(Dautray and Lions [59] and Tychonoff and Samarski [308]). This equation
appears, for instance, in conformal mapping (Gaier [88, 89]), electrostatics
(Gauss [95], Martensen [199] and Stratton [298]), stationary heat conduction
(G¨unter [113]), in plane elasticity as the membrane state and the torsion
problem (Szabo [300]), in Darcy flow through porous media (Bear [12] and
Liggett and Liu [188]) and in potential flow (Glauert [102], Hess and Smith
[124], Jameson [147] and Lamb [181]), to mention a few.
The approach here is based on the relation between the Cauchy data
of solutions via the Calder´on projector. As will be seen, the corresponding
boundary integral equations may have eigensolutions in spite of the unique-
ness of the solutions of the original boundary value problems. By appropriate
modifications of the boundary integral equations in terms of these eigenso-
lutions, the uniquness of the boundary integral equations can be achieved.
Although these simple, classical model problems are well known, the concepts
and procedures outlined here will be applied in the same manner for more
general cases.

1.1 The Green Representation Formula

For the sake of simplicity, let us first consider, as a model problem, the
Laplacian in two and three dimensions. As usual, we use x =(x 1 ,...,x n ) ∈