Lets assume that we want to solve some boundary value problem for the Laplace equation

**Equation 1.6. **

$\phantom{\rule{1.00em}{0ex}}-\Delta \phi =0\phantom{\rule{1.00em}{0ex}}\text{in}\phantom{\rule{0.278em}{0ex}}\Omega \subset {\mathbb{R}}^{3},\phantom{\rule{2.00em}{0ex}}+\text{boundary}\phantom{\rule{0.278em}{0ex}}\text{conditions}\phantom{\rule{0.278em}{0ex}}.$

We could easily plug Equation 1.6 into Green's second identity by what the
domain integral's first term instantly vanishes. For the remaining
domain integral we introduce a two point function
$U(x-\tilde{x})=\psi \left(x\right)$
for which we demand that

**Equation 1.7. **

$\phantom{\rule{1.00em}{0ex}}-{\displaystyle \underset{{\mathbb{R}}^{3}}{\int}}\Delta U(x-\tilde{x})\phantom{\rule{0.167em}{0ex}}\varphi \left(x\right)dx{\displaystyle \stackrel{!}{=}}\varphi \left(\tilde{x}\right)$

holds for
$x\ne \tilde{x}$
. A function with the above property is commonly denoted as
*fundamental solution*. It usually builds the
backbone for every Boundary Element Method since it is its main
ingredient. For the Laplace operator the fundamental solution is simply

**Equation 1.8. **

$\phantom{\rule{1.00em}{0ex}}U\left(z\right)=\genfrac{}{}{0.1ex}{}{1}{4\pi}\genfrac{}{}{0.1ex}{}{1}{\left|z\right|}\phantom{\rule{1.00em}{0ex}}\forall \phantom{\rule{0.167em}{0ex}}z\in {\mathbb{R}}^{3}\phantom{\rule{0.278em}{0ex}}.$

Finally, inserting the fundamental solution into Green's second identity we end up with

**Equation 1.9. **

$\phantom{\rule{1.00em}{0ex}}\phi \left(\tilde{x}\right)={\displaystyle \underset{\partial \Omega}{\int}}\genfrac{}{}{0.1ex}{}{\partial \phi}{\partial \mathbf{n}\left(y\right)}\phantom{\rule{0.167em}{0ex}}U(y-\tilde{x})-\phi \phantom{\rule{0.167em}{0ex}}\genfrac{}{}{0.1ex}{}{\partial}{\partial \mathbf{n}\left(y\right)}U(y-\tilde{x})\phantom{\rule{0.167em}{0ex}}d{s}_{y}\phantom{\rule{1.00em}{0ex}}\forall \phantom{\rule{0.167em}{0ex}}\tilde{x}\in {\mathbb{R}}^{3}\setminus \partial \Omega ,\phantom{\rule{0.167em}{0ex}}y\in \partial \Omega $

which is known as *representation formula*. The
representation formula is a boundary integral equation for the PDE Equation 1.6. It determines the solution of the Laplace
equation completely just by the Cauchy data
$\{\phi \left(x\right),\partial \phi \left(x\right)/\partial \mathbf{n}\left(x\right)\}$
. Unfortunately, the complete Cauchy data is hardly known for some given
boundary value problem. It is therefore the main task of Boundary
Element Methods to complete the Cauchy data by calculating the missing
parts of the Dirichlet data and the Neumann data. This computation can
be done via so-called boundary layer potentials. They will be derived in
the following section and, essentially, they are what the BETL is all
about.