Is the Dirichlet's problem solved in full generality? i.e let $M,N$ be compact oriented Riemannian manifolds with boundary, and let $\psi:\partial M \to N$ be smooth.
Is there always a harmonic map $\phi:M \to N$ which agrees with $\psi$ on $ \partial M$? Are there some sufficient conditions which ensure solutions exist?
Somehow, I did not find a definitive reference. Any help would be appreciated.
Edit:
As explained below there are some topological and differential-topological obstructions. So, suppose $\psi$ is the restriction of a smooth orientation-preserving immersion $M \to N$.
Even in this case, harmonic maps do not always exist:
Take $M=[0,1]$, $N$ a flat annulus in the Euclidean plane. Then, harmonic maps $M \to N$ are geodesics. Choose $\psi(0),\psi(1)$ to be two points such that no geodesic exist between them (the geodesics in $N$ are standard straight segments in $\mathbb{R}^2$ and $N$ is not convex, so just pick two points s.t the connecting segment between them does not lie entirely in $N$ ).
Obstructions:
As commented by levap, there are topological obstructions.
For example, if $M$ is a two-dimensional disc and $N=\mathbb{S}^1$ then a map $\psi:\mathbb{S}^1 \to \mathbb{S}^1$ must be null-homotopic in order to admit a continuous extension to $M$.
Moreover, existence of a continuous extension does not always ensure existence of a smooth extension, see here .