# Poisson's Equation

(Difference between revisions)
To avoid confusion I want to make clear explicitly that we need to distinguish between in/homogenious PDE's and in/homogenious boundary conditions. Inhomogenious PDE's are given by $\mathcal{L}u=\rho$, whereas homogenious PDE's have $\rho=0$. Homogenious boundary conditions mean, that there are essentially no boundary conditions, in electrostatics these are problems with e.g. charges in free space. Inhomogenious boundary conditions are those where additionally to a given charge distribution in space, the solution is constrained to have a certain potential on some region $S$ in space. For example a charge distribution is kept in some metal sphere, which itself is held at a certain potential. We distinguish two boundary-value problems
• Dirichlet boundary conditions, where $u(\mathbf{r})$ is specified on some surface $S$ (e.g. the metal sphere)
• Neumann boundary conditions, where $\partial u(\mathbf{r})/ \partial n$ is given on some surface $S$
Find the solution $G(\mathbf{r},\mathbf{r}_0)$ for Poisson's equation (inhomogenious Laplace's equation) with homogenious boundary conditions in three dimensions that tends to zero as $|\mathbf{r}|\rightarrow \infty$. \begin{align} \nabla ^2 G(\mathbf{r},\mathbf{r}_0)=\delta(\mathbf{r}-\mathbf{r}_0) \end{align}