The usual procedure to solve the Laplace equation in 3 dimensional space is to switch to spherical coordinates, use the seperation of variables approach, which eventually leads to the well-known spherical harmonics. But how do we know that we have found all possible solutions by that procedure?
By using the seperation of variables approach, we made the assumption of $$f(r,\theta,\phi) = f_1(r)f_2(\theta)f_3(\phi).$$ How do we know that there is not a function $f^*$ that does not meet this assumption but still is a solution to the laplace equation in 3d?
Additionally, do not the spherical harmonics form a complete orthogonal base in the space of square-integrable functions? Therefore actually "all" square-integrable functions are solutions of the Laplace equation in 3d?