I am a mathematics student trying to understand physics behind the Poisson equation and the Dirichlet problem. We know that the electric potential potential $u$ satisfies the Poisson Equation:
$\Delta u= -\rho$ (up to a constant)
where $\rho$ is the charge density of some total charge Q contained in some body (i.e. region) $\Omega$.
I guess the potential is some abstract quantity defined at any point in space, could be both outside or inside the body. The density is obviously zero outside the body, so for regions with no charge we have the Laplace equation:
$\Delta u = 0$
Moreover, for a body in electrostatic equilibrium, we know 2 things:
1) it has constant potential inside
2) and all of the charge is located on the boundary
Questions:
1) How is the Dirichlet's problem related to all this? I initially thought that since there is no charge inside the body, we have $\Delta u = 0$ on $\Omega$, but it is not clear to me how we can get $u = f$ on $\partial \Omega$ and what should f be. Clearly f is not equal to the value of potential on $\partial \Omega$ which is constant on the surface. Then what is f?
2) If for any body the density is $\textbf{always}$ zero inside and outside the body, then the Poisson equation for the potential always reduces to Laplace equation inside or outside the body. $\rho$ is only non-zero on the boundary. Then where do we encounter the Poisson equation as it is, with $\Delta u(x) = g(x)$ for $x\in \Omega$, not $x\in\partial\Omega$?
References for books are much appreciated. Thanks
The Wikipedia article on Newtonial potential addresses the electrostatic potential of a charged body at the end: the function $u$ is a single layer potential. It satisfies the Laplace equation both in the interior (trivially, by being constant) and in the exterior (which is of interest). So, one usually considers this as a Dirichlet problem in the exterior domain, that is in $G = \mathbb{R}^3\setminus \overline{\Omega}$. So the problem is then to find a harmonic function in $G$ that is equal to a constant $C$ on the boundary $\partial G$ and tends to $0$ at infinity. (This is for three and more dimensions; in two dimensions the potential must have logarithmic growth at infinity instead of decay.) The value of $C$ depends on how much charge is put on the body $\Omega$; it is basically charge divided by the capacity of $\Omega$.
The Poisson equation models a charge distributed across some three-dimensional region. I imagine it as charged dust particles suspended in the air, which perhaps doesn't happen often (search for "3d charge density" or "volume charge density" comes up mostly with textbook exercises). But it has a natural interpretation as the gravitational potential of a gas cloud.