We establish the existence of locally minimizing vortex solutions to the reduced and full Ginzburg-Landau energy in three dimensional simply-connected domains with or without the presence of an applied magnetic field. The approach is based upon the theory of weak Jacobians and applies to nonconvex sample geometries for which there exists a configuration of locally shortest line segments with endpoints on the boundary. This is joint work with Robert Jerrard, Alberto Montero and William Ziemer.