Symmetric vortices for two-component Ginzburg-Landau systems

We study Ginzburg--Landau equations for a complex vector order parameter Psi=(psi_+,psi_-). We consider symmetric (equivariant) vortex solutions in the plane R^2 with given degrees n_\pm, and prove existence, uniqueness, and asymptotic behavior of solutions for large r. We also consider the monotonicity properties of solutions, and exhibit parameter ranges in which both vortex profiles |psi_+|, |psi_i| are monotone, as well as parameter regimes where one component is non-monotone. The qualitative results are obtained by means of a sub- and supersolution construction and a comparison theorem for elliptic systems.Comment: 32 page

Gamma-convergence of 2D Ginzburg-Landau functionals with vortex concentration along curves

We study the variational convergence of a family of two-dimensional Ginzburg-Landau functionals arising in the study of superfluidity or thin-film superconductivity, as the Ginzburg-Landau parameter epsilon tends to 0. In this regime and for large enough applied rotations (for superfluids) or magnetic fields (for superconductors), the minimizers acquire quantized point singularities (vortices). We focus on situations in which an unbounded number of vortices accumulate along a prescribed Jordan curve or a simple arc in the domain. This is known to occur in a circular annulus under uniform rotation, or in a simply connected domain with an appropriately chosen rotational vector field. We prove that, suitably normalized, the energy functionals Gamma-converge to a classical energy from potential theory. Applied to global minimizers, our results describe the limiting distribution of vortices along the curve in terms of Green equilibrium measures

Minimizers of the Landau-de Gennes energy around a spherical colloid particle

We consider energy minimizing configurations of a nematic liquid crystal around a spherical colloid particle, in the context of the Landau-de Gennes model. The nematic is assumed to occupy the exterior of a ball of radius r_0, satisfy homeotropic weak anchoring at the surface of the colloid, and approach a uniform uniaxial state at infinity. We study the minimizers in two different limiting regimes: for balls which are small compared to the characteristic length scale r_0>L. The relationship between the radius and the anchoring strength W is also relevant. For small balls we obtain a limiting quadrupolar configuration, with a "Saturn ring" defect for relatively strong anchoring, corresponding to an exchange of eigenvalues of the Q-tensor. In the limit of very large balls we obtain an axisymmetric minimizer of the Oseen-Frank energy, and a dipole configuration with exactly one point defect is obtained

Sharp interface limit of an energy modelling nanoparticle-polymer blends

We identify the $\Gamma$-limit of a nanoparticle-polymer model as the number of particles goes to infinity and as the size of the particles and the phase transition thickness of the polymer phases approach zero. The limiting energy consists of two terms: the perimeter of the interface separating the phases and a penalization term related to the density distribution of the infinitely many small nanoparticles. We prove that local minimizers of the limiting energy admit regular phase boundaries and derive necessary conditions of local minimality via the first variation. Finally we discuss possible critical and minimizing patterns in two dimensions and how these patterns vary from global minimizers of the purely local isoperimetric problem.Comment: Minor changes. Rephrased introduction. This version is to appear in Interfaces and Free Boundarie

Thin film limits for Ginzburg--Landau with strong applied magnetic fields

In this work, we study thin-film limits of the full three-dimensional Ginzburg-Landau model for a superconductor in an applied magnetic field oriented obliquely to the film surface. We obtain Gamma-convergence results in several regimes, determined by the asymptotic ratio between the magnitude of the parallel applied magnetic field and the thickness of the film. Depending on the regime, we show that there may be a decrease in the density of Cooper pairs. We also show that in the case of variable thickness of the film, its geometry will affect the effective applied magnetic field, thus influencing the position of vortices