$\Delta_g f = 0$ on the Riemannian Manifold $(\mathbb{R}^3 \setminus B , g)$ with conditions on the boundary and at infinity

Question

Consider the manifold $\mathbb{R}^3 \setminus B$ where $B$ is the ball with radius 1 with riemannian metric $g$ (not necessarily the euclidean metric).

I am looking for solutions to $\Delta_g f = 0$ where $f \to 1$ at infinity and $f=f_0$ on $\partial M$ where $f_0$ is some positive function on $∂M$.

1) What can we say about existence and uniqueness of this PDE? Any simple proofs?

2) What can we say about the asymptotic behaviour of f at infinity?

I think if g is the euclidean metric, the asymptotics are: $$f = 1+ \frac{C}{r} + O \left( \frac{1}{r^{2}} \right)$$ where $C$ is some constant and $r=\sqrt{x^2+y^2+z^2}$ in cartesian coordinates. (Check chapter 2.17 in The Laplace Equation by Dagmar Medkova).

3) What if g is asymptotically flat? (so in cartesian coordinates, the metric satisfies $ g_{ij} = \delta_{ij} + O \left( \frac{1}{r^{\delta}} \right) $ for some $\delta > 0$).

Any help is appreciated. If you know any references, please share it with me.

Answer 1

You will find a full treatement of this problem in: The Mass of Asymptotically Flat Manifold, by Bartnik. Becarefull there is a small error, the decay rate of $f-1$ can't be better than $\frac{1}{r}$, that is to say the one of the Green function.