  In this paper, we solve the Dirichlet problem for Sobolev maps between
singular metric spaces that extends the corresponding result of Guo and Wenger
[Comm. Anal. Geom. 2020]. The main new ingredient in our proofs is a suitable
extension of the theory of trace for metric valued Sobolev maps developed by
Korevaar and Schoen [Comm. Anal. Geom. 1993]. We also develop a theory of trace
in the borderline case, which investigates a sharp condition to characterize
the existence of traces.
