  We study the Dirichlet problem for minimal surface systems in arbitrary
dimension and codimension via mean curvature flow, and obtain the existence of
minimal graphs over arbitrary mean convex bounded $C^2$ domains for a large
class of prescribed boundary data. This result can be seen as a natural
generalization of the classical sharp criterion for solvability of the minimal
surface equation by Jenkins-Serrin. In contrast, we also construct a class of
prescribed boundary data on just mean convex domains for which the Dirichlet
problem in codimension 2 is not solvable. Moreover, we study existence and the
uniqueness of minimal graphs by perturbation.
