Suppose you have an harmonic function on an exterior domain of $\mathbb{R}^n$, i.e., a function $u \colon \mathbb{R}^n \setminus \bar\Omega \to \mathbb{R}$, where $\Omega$ is a smooth and bounded open set and $u$ satisfies $$ \left\{ \begin{array}{ll} \Delta u = 0 & \text{in $\mathbb{R}^n \setminus \bar\Omega$} \\ u = 1 & \text{on $\partial\Omega$} \\ u(x) \to 0 & \text{for $x \to \infty$} . \end{array} \right. $$ It seems sensible to me that level sets $\{u=t\}$ of $u$ are definitely diffeomorphic to spheres for small $t$. I expect $\nabla u$ not to vanish for small $t$ (and I can assume that if it is not obvious) and thus the level set flow of $u$ should provide some sort of identification between $\{u=t\}$ and the "sphere at infinity". But I am not able to prove this. Can someone help? Is this a known fact?
EDIT I need only the case $n\geq3$.
The function $u$ is the Newtonian potential of the equilibrium measure compact set $\overline{\Omega}$ (I refer to Markina's notes for definitions). That is, there is a positive measure $\mu$ supported on $\overline{\Omega}$ such that $$ u(x) = \int_{\overline{\Omega}} |x-y|^{2-n}\,d\mu(y) \tag{1} $$
This can also be seen more directly by observing that $u$ is a superharmonic function on $\mathbb R^n$, writing out its Riesz decomposition $u=|x|^{2-n}* \mu +h$ (where $h$ is a harmonic function on $\mathbb{R}^n$) and observing that $h\equiv 0$ due to vanishing at infinity.
Differentiating (1) for $x\notin \overline{\Omega}$ yields $$ \nabla u(x) = (2-n)\int_{\overline{\Omega}} \frac{x-y}{|x-y|^{n}}\,d\mu(y) \tag{2} $$ When $|x|$ is sufficiently large, all vectors $x-y$ point in about the same direction as $x$; there is no cancellation between them. This implies $\nabla u(x)\ne 0$ and moreover, that the level surface $\{u=t\}$ bounds a star-shaped domain when $t$ is small. More specifically, the position vector $x$ is transverse to the surface, since moving from a point on this surface toward $0$ increases $u$ at a positive rate.
Conclusion: the radial projection of $\{u=t\}$ onto the sphere $|x|=1$ is a diffeomorphism when $t$ is small enough.