I am looking some references on the existence and non-existence of (space-like) harmonic maps solving the Dirichlet into the de-Sitter space.
More precisely: Let, for $n\geq 3$, $$dS^n=\{ u\in \mathbb{R}^{n,1} \, \vert \, \langle u,u\rangle =1 \}$$ where
$$\langle .,.\rangle= -dx_0^2 +\sum_{i=1}^n dx_i^2 .$$
Given $\phi : S^1 \rightarrow dS^n$ (smooth), we look for an harmonic map $$u:\mathbb{D} \rightarrow dS^n$$
such that
$$u_{\partial \mathbb{D}} =\phi.$$
As in the Euclidean(and Riemannian) case, one idea is to proceed by minimization, but since the metric is not positive, there is no chance for this method to work.
One idea is to assume that $\phi$ can be bounded by a disc in de Sitter whose tangent planes are space-like and then to minimize only among the space-like map: $$\{ u:\mathbb{D} \rightarrow dS^n \, \vert \, \vert \nabla u\vert^2\geq 0 \}.$$
This will provide an inferior bound on the functional and give some some chance to a minimization process to work. Unfortunately I found nothing in this spirit in the Literature. I will be grateful for any result reference on existence, uniqueness, etc...