First uniqueness theorem:

The solution to Laplace’s equation in some volume \(\mathscr{V}\) is uniquely determined if the \(V\) (the potential) is specified on the boundary surface \(S\).

Second uniqueness theorem:

In a volume \(\mathscr{V}\) surrounded by conductors and containing a specified charge density \(\rho\), the electric field is uniquely determined if the total charge on each conductor is given.