**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.



Harmonic functions in infinite domain in Euclidean space
Assuming "vanish at infinity" means that



> 
> for every $\epsilon > 0$ there exists a sufficiently large ball $B\_\epsilon$ such that $\big|f|\_{\mathbb{R}^n \setminus B\_\epsilon}\big| < \epsilon$
> 
> 
> 


then you can just apply the maximum principle to $B\_\epsilon \setminus \Omega$ and conclude that $f$ is bounded by $\epsilon$ on that set.


Take $\epsilon \to 0$ you get that $f$ must vanish identically.


No decay rate needed.


