  In this note, a brief introduction to the physical and mathematical
background of the two-component Ginzburg-Landau theory is given. From this
theory we derive a boundary value problem whose solution can be obtained in
part by solving a minimization problem using the technique of variational
method, except that two of the eight boundary conditions cannot be satisfied.
To overcome the difficulty of recovering the full set of boundary conditions,
we employ a variety of methods, including the uniform estimation method and the
bounded monotonic theorem, which may be applied to other complicated vortex
problems in gauge field theories. The twisted vortex solutions are obtained as
energy-minimizing cylindrically symmetric field configurations. We also give
the sharp asymptotic estimates for the twisted vortex solutions at the origin
and infinity.
