General question about solutions to Laplace's equation

[AxiomOfChoice]

Is it true that any solution to Laplace's equation, subject to any set of boundary conditions, can be written as a linear superposition of separable solutions?

I'm sure there are some vagaries in what I've written above. Feel free to point them out and rectify them.

 

[arunma]

I believe there's some theorem in math which says that any function can be written as a linear combination of orthogonal functions. And I'm pretty sure that any set of boundary conditions on solutions to Laplace's Equation uniquely defines one solution. If you've derived a family of separable solutions to Laplace's Equation, then it must consist of functions with some sort of orthogonality relation (e.g. Legendre Polynomials, spherical harmonics, trigonometric functions), and so you can write any function you like as a linear combination of them. So I don't see why what you're saying shouldn't be true.

 

[HallsofIvy]

No, I believe that there exist "geometries" (shapes of the boundary) for which Laplace's equation is NOT separable. Unfortunately, I can't think of any off hand.