We study solutions of 2D Ginzburg-Landau (GL) equation for a complex valued order parameter $u$
$$
-\Delta u+\frac{1}{\varepsilon^2}u(|u|^2-1)=0, \, \, x \in A \subset R^2
$$
which is lies in the core of Ginzburg-Landau theory of superconductivity and superfluidity.
For a 2D domain $A$ with holes we consider the so-called "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 that there are
stable solutions of this problem with vortices. These are vortices of a novel type: they approach the boundary
and have bounded energy in the limit of small $\varepsilon$. In the well studied case of the Dirichlet problem for GL
PDE stable solutions contain vortices distant from boundary with energy that blows up as $\varepsilon \to 0$.
% For the
%Dirichlet boundary condition ("stiff" problem) the existence of stable solutions with vortices whose energy blows up as
$\varepsilon \to 0$ is well known. These vortices are distant from the boundary.
By contrast, in the
case of homogeneous Neumann ("soft") problem stable solutions with vortices are not established.
In this work we develop variational method which allows to construct
local minimizers of the corresponding Ginzburg-Landau
energy functional. We introduce approximate bulk degree which is the key
ingredient of this method, and it is preserved in the weak $H^1$-limit
unlike the standard degree over a curve.
This is a joint work with V. Rybalko. |