  We study solutions of the 2D Ginzburg-Landau equation -\Delta
u+\frac{1}{\ve^2}u(|u|^2-1)=0 subject to "semi-stiff" boundary conditions: the
Dirichlet condition for the modulus, |u|=1, and the homogeneous Neumann
condition for the phase. The principal result of this work shows there are
stable solutions of this problem with zeros (vortices), which are located near
the boundary and have bounded energy in the limit of small epsilon. For the
Dirichlet bondary condition ("stiff" problem), the existence of stable
solutions with vortices, whose energy blows up as epsilon goes to 0, is well
known. By contrast, stable solutions with vortices are not established in the
case of the homogeneous Neumann ("soft") boundary condition. (nonexistence is
proved for simply connected domains).
  In this work, we develop a variational method which allows one to construct
local minimizers of the corresponding Ginzburg-Landau energy functional. We
introduce an approximate bulk degree as the key ingredient of this method, and,
unlike the standard degree over the curve, it is preserved in the weak
H^1-limit.
