Find the electric potential $\phi$, satisfying $\nabla^2 \phi=0$ between the two cylinders $r=a$, on which $\phi=0$, and $r=b>a$, on which $\phi=V$. Suppose that the inner cylinder is perturbed to $r=a(1+\epsilon \sin n\theta)$. Calculate $\phi$ correct to $O(\epsilon)$; to build up your arithmetical strength, calculate it correct to $O(\epsilon^2)$. What restriction on $n$ is necessary for your expansion to be valid?
This is a domain perturbation problem. The difficulty here is the domain rather than the coefficients of a PDE is perturbed.
We know the solution when $\epsilon=0$, it is a harmonic function on a ring. Then how to deal with the perturbation?