## Fundamental solution and representation formula

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

Equation 1.6.

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

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 $U\left(x-\stackrel{˜}{x}\right)=\psi \left(x\right)$ for which we demand that

Equation 1.7.

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

holds for $x\ne \stackrel{˜}{x}$ . 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.

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

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

Equation 1.9.

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

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 $\left\{\phi \left(x\right),\partial \phi \left(x\right)/\partial \mathbf{n}\left(x\right)\right\}$ . 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.