We study solutions of the 2D Ginzburg–Landau equation $-\Delta u+{\epsilon }^{-2}{u\left(\right|u|}^2-1)=0$ subject to “semi-stiff” boundary conditions: Dirichlet conditions for the modulus, $\left|u\right|=1$, and homogeneous Neumann conditions for the phase. The principal result of this work shows that 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 boundary condition (“stiff” problem), the existence of stable solutions with vortices, whose energy...

On étudie la fonctionnelle d’énergie de Ginzburg-Landau$$J(u,A)=\frac{1}{2}{\int }_{\Omega }|{\nabla }_A{u|}^2+|h-h_{ex}{|}^2+\frac{{\kappa }^2}{2}{(1-|u|}^2{)}^2,$$qui modélise les supraconducteurs cylindriques soumis à un champ magnétique extérieur $h_{ex}$, dans l’asymptotique $\kappa \to \infty$. On trouve et on décrit des branches de solutions stables des équations associées. On a une estimation sur la valeur critique $H_{c_1}\left(\kappa \right)$ de $h_{ex}$ correspondant à une «transition de phase» où des vortex (c.à.d. zéros de $u$) deviennent énergétiquement favorables. On obtient également dans le cas d’un disque, que pour $h_{ex}\le H_{c_1}$ comme pour $h_{ex}\ge H_{c_1}$, il existe à la...

We classify nonconstant entire local minimizers of the standard Ginzburg–Landau functional for maps in $H_{\text{loc}}^1({ℝ}^3;{ℝ}^3)$ satisfying a natural energy bound. Up to translations and rotations,such solutions of the Ginzburg–Landau system are given by an explicit solution equivariant under the action of the orthogonal group.

Systems with Coulomb and logarithmic interactions arise in various settings: an instance is the classical Coulomb gas which in some cases happens to be a random matrix ensemble, another is vortices in the Ginzburg-Landau model of superconductivity, where one observes in certain regimes the emergence of densely packed point vortices forming perfect triangular lattice patterns named Abrikosov lattices, a third is the study of Fekete points which arise in approximation theory. In this review, we describe...