Abstract
We consider the location of near boundary vortices which arise in the study of minimizing sequences of Ginzburg-Landau functional with degree boundary condition. As the problem is not well-posed --- minimizers do not exist, we consider a regularized problem which corresponds physically to the presence of a superconducting layer at the boundary. The study of this formulation in which minimizers now do exist, is linked to the analysis of a version of renormalized energy. As the layer width decreases to zero, we show that the vortices of any minimizer converge to a point of the boundary with maximum curvature. This appears to be the first such result for complex-valued Ginzburg-Landau type problems.
Highlights
We consider the Ginzburg-Landau (GL) functional (1) that has been the subject of many studies since the celebrated monograph [4]: E(u) =
U is a complex-valued function, Ω is for simplicity, a bounded, smooth and connected domain in R2. This functional arises in the studies of superconductivity. It is a simplified version of the physical model due to the absence of magnetic field in the functional
It is striking to see that the degree deg(g, ∂Ω) on the boundary condition creates the same quantized vortices as a magnetic field in type-II superconductors or as an
Summary
The above setting, though stated, already leads to some interesting phenomena It was shown in [6] that any minimizing sequence {uk}k≥1 of (1) in class (2) develops vortices {zk}k≥1 which approach the boundary ∂Ω as k −→ ∞. A reasonable question is to ask, “What is the best rate of convergent of the energy to the minimum value as a function of the distance of the vortex to the boundary?” This heuristic concept can be formalized as follows: for any given number ρ > 0, find a function uρ which minimizes (1) from class (2) with an additional constraint that uρ has d vortices but with distances of at least ρ from the boundary ∂Ω.
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