We study solutions of the Ginzburg-Landau (GL) equation for a complex valued order parameter $u$

\begin{displaymath} -\Delta u+\frac{1}{\varepsilon2}u(\vert u\vert^2-1)=0, x \in A \subset R2. \end{displaymath}

This equation is of principal importance in the Ginzburg-Landau theory of superconductivity and superfluidity.

For a 2D domain $A$ with holes we consider the so-called ``semi-stiff'' boundary conditions: the the Dirichlet condition for the modulus $\vert u\vert=1$, and the homogeneous Neumann condition for the phase $\arg(u)$. The principal result of this work is that there are stable solutions with vortices of this boundary value problem. The vortices are of a novel type: they approach the boundary and have bounded energy in the limit of small $\varepsilon$. By contrast, in the well-studied Dirichlet problem for the GL PDE, the vortices are distant from boundary and their energy blows up as $\varepsilon \to 0$. Also, the existence of stable solutions to the homogeneous Neumann (``soft'') problem with vortices has not been established.

In this work we develop a variational method that allows us to construct local minimizers of the GL energy functional which corresponds to the GL PDE. We introduce the notion of the approximate bulk degree which is the key ingredient of our method. We show that, unlike the standard degree over a curve, the approximate bulk degree is preserved in the weak $H1$-limit.

This is a joint work with V. Rybalko.