Solutions with vortices of a semi-stiff boundary value problem for the Ginzburg–Landau equation

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