# Boundary conditions in Poisson's equation for gravity

Say we want to calculate the gravitational potential everywhere around(outside) a solid, circular, right cylinder. We want to use Poisson's equation for gravity for that (Laplace(U) = -4*pi*density or something like that). Since we want to know the potential outside the cyllinder this reduces to laplaces equation laplace(U) = 0

My question is: What would the boundary conditions for this equation be? Obviously one is that it decays to zero at infinity, but what about the boundary conditions on surface of the cyllinder?

The boundary condition for this equation can be whatever you want, Dirichlet (value of $U({x})$) or Neumann (value of $\nabla U({x}).n(x)$), but you need first to compute it via the integral $$U({y}) = -\int_{cylinder} \frac{G\,\rho(x)}{|{y}-{x}|}\ dx$$ where $\rho({x})$ is the density inside your cylinder.
Assuming there is no infinitely thin surface density, the boundary conditions are continuity of the potential and continuity of the derivative of the potential: $$U_{in} = U_{out}, \qquad \frac{\partial U_{in}}{\partial n} = \frac{\partial U_{out}}{\partial n},$$ at the surface, where $n$ is the normal coordinate to the surface. These are due to the gravitational force being finite and continuous.