**EDIT**: Let $\Omega\subset \mathbb{R}^n$ be a bounded domain with smooth boundary.  Let $f\colon \mathbb{R}^n\backslash \Omega \to \mathbb{R}$ be a continuous function which is harmonic in $\mathbb{R}^n\backslash \bar\Omega$ and vanishes on $\partial \Omega$. Let us also assume that $f$ vanishes at infinity. 

**What assumptions on the decay rate at infinity imply that $f\equiv 0$?**

The case $n=3$ is of special interest to me.

**ADD:** The motivation of my question comes from the very classical problem from electrostatics ($n=3$) which is probably solved. Assume the domain $\Omega$ is filled with a conductor and electrified with a charge. All the charged is necessarily accumulated on the surface of the domain. The potential of the created electric field in the space is a harmonic function outside if the domain and is constant on the boundary. It decays at infinity like $1/r^2$. **Is such potential unique? Equivalently, is distribution of charge on the surface is unique?**

Probably this is a very well studied question, but I am not a specialist.