Solutions to Dirichlet problem on manifolds with boundary I am looking for a reference for the following assertion:
Let $M$ be a Riemannian manifold with boundary, and $f:\partial M \rightarrow \mathbb{R}$ be smooth. Then there exists a unique smooth function $ w:M \rightarrow \mathbb{R} $ such that $w|_{\partial M} = f$ and $ \Delta w = 0$ (where $ \Delta $ is the Laplacian on $M$).
In particular, does existence and uniqueness always hold?  (without any limitations on the curvature of $M$, I have seen sources which treat the special case of negative curvature like "Lectures on  Differential Geomery" by Schoen & Yau)
The case which interests me most is the one when $M$ is a compact ball in $\mathbb{R^n}$ but the metric is arbitrary. (It might be different from the Euclidean)
Update: As mentioned (in a comment by  Chris Gerig), uniqueness does not hold for noncompact manifolds. So lets narrow down the discussion to the compact case.
 A: For ball in $\mathbb{R^n}$ it is easy. You can cover whole ball with one map(for example identity map :D) And write you equation $\Delta w =0$ in coordinates. This will only alter the Laplace operator a little bit and you end up solving equation in to form
$$
\nabla \cdot (A\,\nabla w) = 0
$$
Where
$$
A_{ij} = \sqrt{|g|} g^{ij}
$$
see wiki page on Laplace-Beltrami operator.
So you have standard Dirichlet problem problem in $\mathbb{R^n}$, therefore existence and uniqueness is guaranteed. If you want reference that for example consult Evans book on PDE.

On any compact manifold with boundary. I think that the result should be the same, in the proof existence on $\mathbb{R^n}$ you do not use any global  properties of $\mathbb{R^n}$. I think It should be reasonably easy to alter the existence proof from $\mathbb{R^n}$ to any compact manifold with boundary, but a little bit tedious because you have to develop theory of  Sobolev spaces on that manifold.

On compact manifold without boundary. Here I answer a question that has not been asked but seems to me interesting.
What about equation 
$$
\Delta w = g
$$
Here we have to be careful. Necessary condition on existence is $\int g = 0$. This can be seen from a little physical intuition. This equation can be considered as stationary solution of heat equation
$$ 
\partial_t w - \Delta w = g
$$
In stationary situation there has to be balance between energy inflow and outflow. Without boundary the cannot be any energy flow through it, thus the necessary condition $\int g = 0$. Although I do not know if it is sufficient condition for existence.
But it is probably very similar to the purely Neumann problem, where condition $\int g=0$ is sufficient.
