I have the following problem: $$\Delta u = 0\;in\;\Omega$$ with several boundary conditions.
Applying Green's second identity the representation formula can be derived: $$u(\mathbf{x})=\int_{\Gamma}\left(\partial_nu\;G(\mathbf{x},\cdot)-u\;\partial_nG(\mathbf{x},\cdot)\right)\;\mathrm{d}\Gamma,\;\text{for almost every}\;\mathbf{x}\in\Omega$$ where $G(\mathbf{x},\cdot)$ is the fundamental solution of the laplacian.
Now, it is supposed that if we restrict to $\mathbf{x}\in\Gamma$ where $\Gamma$ is the boundary we get the following equation: $$u(\mathbf{x})=\int_{\Gamma}\partial_nu\;G(\mathbf{x},\cdot)\;\mathrm{d}\Gamma-\int_{\Gamma}u\;\partial_nG(\mathbf{x},\cdot)\;\mathrm{d}\Gamma+\frac{1}{2}u(\mathbf{x}),\;\text{for almost every}\;\mathbf{x}\in\Gamma,$$where there is a jump in the second integral.
My question is that I do not understand how did we get the term $\frac{1}{2}u(\mathbf{x})$. I guess that is something related about the fact that there is a jump in the boundary so as you approximate to the boundary something happens, but I do not understand what.
Just to say that this is done in the boundary element method.
Thanks!
First of all, a more accurate way to state the domain of $\mathbf{u}$ in the first integral equation is that $\mathbf{x} \not \in \Gamma$.
The integral kernel $\partial_{n}G(\mathbf{x},\cdot)$ becomes singular when $\mathbf{x} \in \Gamma$, so the integral is understood in the principal value sense in the second integral equation you have. When $\mathbf{x} \not \in \Gamma$, the integrand is smooth so you don't need to worry about it in the first integral equation.
One can show that for any continuous function $\phi$ defined on $\Gamma$, $$\lim_{\mathbf{x}\rightarrow \mathbf{x}_{0} \in \Gamma}\int_{\Gamma}\phi(\cdot)\partial_{n}G(\mathbf{x},\cdot)d\Gamma =\int_{\Gamma}\phi(\cdot)\partial_{n}G(\mathbf{x}_{0},\cdot)d\Gamma -\frac{1}{2}\phi(\mathbf{x}_{0}) $$ where in the limit $\mathbf{x}$ approaches $\mathbf{x}_{0}$ from the interior of $\Omega$. This is one of the so called "jump relations". To see this, you can use the formula for the Green's function and do some honest calculus or consult books by Rainer Kress and David Colton, for example.
Now take the limit as $\mathbf{x}$ approaches $\Gamma$ in the first integral equation and use the above result you have the second integral equation. You are looking at a limit of $\mathbf{u}$ on $\Gamma$ instead of the restriction because in the first integral equation, $\mathbf{u}$ is not defined on $\Gamma$. And yes $\mathbf{u}$ has different limits when you approach from the interior and exterior of $\Omega$.