Suppose I have a PDE which can be solved by an expansion on the Legendre Polynomial basis, for an axisymmetric problem in spherical coordinates.
For the derivation for such a problem, see these notes, pp. 6/24-7/25)
Given:
$\nabla^2 \Phi(r,\theta) = 0$
The solution is:
$\Phi(r,\theta)=\sum_\limits{n=0}^{\infty}\left(A_n r^n + B_n r^{-(n+1)}\right) P_n (\cos\theta)$
This equation can be and is (frequently) used to solve the homogeneous Laplace and non-homogeneous Poisson equation in this coordinate system and the domain $r \in [r_{min},r_{max}]$ (where $0 \le r_{min} \lt r_{max} \le \infty$) and $\theta \in [0,\pi]$. Uniqueness of the solution and closure relations aren't too difficult for those familiar with this approach to solving PDEs.
My question is this:
What if the domain doesn't reach to the poles? What if instead of $\theta \in [0,\pi]$, the domain for $\theta$ is $\theta \in [\epsilon,\pi-\epsilon]$? Would I still be able to use the same techniques to solve for $\Phi(r,\theta)$ on this reduced domain as I would for the full domain? Would there be any consequences to using a reduced domain, such as new closure or uniqueness?