Asymptotics for the minimization of a Ginzburg-Landau functional

Multi-Vortex Non-radial Solutions to the Magnetic Ginzburg-Landau Equations

“…In the case of the Ginzburg-Landau equation on bounded domains, nonradial non-magnetic solutions have been found by Bethuel, Brezis, and Helein [2,3] and non-radial magnetic solutions have been found by Sandier and Serfaty (see references in [23]). This is due to the boundary forces which keep repelling vortices within the bounded domain.…”

“…Stationary states of superconductors occupying (for simplicity) the plane R 2 , are described by pairs (ψ, A), where ψ : R 2 → C is the order parameter and A : R 2 → R 2 is the magnetic potential. These states satisfy the system of equations −∆ A ψ + λ(|ψ| 2 The Ginzburg-Landau equations on the plane model superconductors in R 3 which are spatially homogeneous in one direction, when neglecting boundary effects [14]. They also describe equilibrium states of the U (1) Higgs model of particle physics [16].…”

Uniqueness of vortexless Ginzburg-Landau type minimizers in two dimensions

“…2 It is common knowledge in the GL community that |u| close to 1 plus global pointwise estimates lead to uniqueness. 3 Thus M depends on Ω, on ε and on the degrees.…”

“…of zeros. The usual arguments leading to clearing out make use of the global estimate |∇u| ≤ C ε , 0 < ε < 1, valid if the boundary datum g is smooth and fixed [3]. Such an estimate need not hold if g is merely H 1/2 .…”

“…3 There is some explicit universal constant δ such that, if M < δ, then the infimum is attained and the minimizer is unique (modulo S 1 ).…”