Lower Bounds for Generalized Ginzburg--Landau Functionals

Abstract

Abstract. We study properties of Ginzburg-Landau functionals I U (·), defined for functions u ∈ W 1,n (U ; R n ), where U ⊂ R n . In particular, we establish lower bounds relating the energy I U (u) to the Brouwer degree of u, and we prove under additional hypotheses that the energy concentrates on a small number of small sets. As a consequence we deduce some compactness theorems. Such estimates are useful in studying Ginzburg-Landau-type PDEs associated with the functional I U .

Vortex Liquids and the Ginzburg–landau Equation

“…This u is referred to as the canonical harmonic map when ψ is a harmonic function. This limiting behavior has been established in many situations; see, for example, [3,31,40,19,20]. When dynamics (1) are turned on, these vortices move according to the gradient flow of the Kirchhoff-Onsager energy:…”

The crystallization conjecture: a review

“…Under some constraints on the magnetic field, there is a finite number of vortices which behave like classical particles interacting via the (two-dimensional) Coulomb potential and submitted to a harmonic confining potential [212,134,222,221,213]. For extremely intense magnetic fields, the number of vortices tends to infinity, and the limit problem becomes that of Wigner's crystallization (see Section 2.6), as shown in [214].…”

Mean field limits for Ginzburg-Landau vortices

“…• the vortex ball construction [J1,Sa] which allows to bound the energy of the vortices from below in disjoint vortex balls B i by π|d i || log ε| and deduce that the energy outside of ∪ i B i is controlled by the excess energy E ε − πN ε | log ε|…”