Equation of distribution of electric charge on surface of conductor

In all standard courses of electrostatics there is a discussion of various special cases of the problem of distribution of electric charge on the surface of a conductor when there is given a distribution of charges outside of the conductor. However I have never seen a general equation for the surface charge density (or potential).

What is the equation for the surface charge density or potential of electric field? Does this equation have a unique solution (under reasonable assumptions on regularity on the boundary of the conductor)?

I ask this question on this mathematical site since I believe that this question became closer to mathematics rather than physics.