Conformal mapping makes it possible to exactly solve the 2D Laplace's equation for great many domains and boundary conditions.
For example, I transformed a lens shaped domain (see the figure) to an angle shaped one and got an exact solution to Laplace's equation in 2D with step boundary conditions:
$$\frac{\partial ^2 f(x,y)}{\partial x^2}+\frac{\partial ^2 f(x,y)}{\partial y^2}=0$$
$$f(S_1)=1 \\ f(S_2)=0$$
I don't remember the formula, the point is, I've got an exact explicit solution $f(x,y)$.
But I actually need to solve a similar problem in 3D, with $x,y$ being replaced by $\rho, z$, i.e. the drop lying on a flat surface.
I found no exact solution for this axial symmetric problem, even using an infinite series based on Legendre polynomials.
The problem is, the domain is a mix of spherical symmetry and cylindrical symmetry. Either of these two coordinate systems can't deal with both boundaries at the same time.
So, I would like to know, if there is some way to approximate the solution to the axial symmetric 3D Laplace equation using the known exact solution to the 2D problem with the same geometry (if we exchange the cylindrical coordinates $\rho,z$ to the cartesian ones $x,y$)?