I want to solve a PDE where I know the solutions for one boundary condition and need the solution for a slightly perturbed boundary.
As an example, suppose I know the solution to Laplace's equation $\nabla ^2 V = 0$ where $V=0$ on $f(x,y,z)=0$ and want to find the solution for $V=0$ on $f(x,y,z)+g(x,y,z)=0$ where $g\ll f$. I can't find a reference for how to find a perturbative solution to this.
(My particular problem is that I want to find the potential $V$ where $V=0$ on the surface $r=1+\delta \sin{\phi}$ in the usual spherical polar coordinates and $\delta\ll 1$.)