A boundary value problem related to the Ginzburg-Landau model

Abstract

We analyze the Ginzburg-Landau equation for a superconductor in the case of a 2-dimensional model: a cylindrical conductor with a magnetic field parallel to the axis. This amounts to find the extrema of the free energywhere Ω is a bounded domain with smooth boundary in IR 2 , A = (A ί9 A 2 ) the vector potential, B A = d 1 A 2 -d 2 A ί the magnetic field, Φ Ά complex field. We describe the connected components of the maximal configuration space, i.e. of the set of all {A, Φ) with components in the Sobolev space H ί (Ω) and such that | Φ\ = 1 on the boundary, modulo the action of the gauge group. In the critical case K = 1 we give a complete description of the minimal configurations in each component.

Minimax Critical Points in Ginzburg-Landau Problems with Semi-Stiff Boundary Conditions: Existence and Bubbling

“…15 Then lim |z|→1 |u(z)| = 1 uniformly ⇐⇒ u is a Blaschke product. 13 The first convergence is obtained by combining the fact that f n → f in H 1/2 with dominated convergence.…”

“…The case d < 0 is obtained from the case d > 0 by complex conjugation. 15 We denote by Hol (Ω) the class of holomorphic functions in Ω.…”

“…If u ∈ E and we let g = tr u, then g ∈ H 1/2 (Γ; S 1 ), and therefore we may define the winding number (degree) of g [15,Appendix], denoted by deg (u, Γ) or deg (g, Γ).…”

“…We follow mainly unpublished lecture notes of a graduate course of H. Brezis at Paris 6. Some of the results in this section are also presented in [15,20,21,14,19]. Proofs of Lemmas 2.1-2.3 and 2.5-2.8 are part of Ginzburg-Landau folklore, but most of them unavailable; see however [37, …”

One cannot hear the winding number

“…The earliest investigations in this direction are due to L. Boutet de Monvel and O. Gabber. They realized that the left-most equality in (1) makes sense for functions in the fractional Sobolev space W 1/2,2 , when the integral is understood in the sense of W 1/2,2 -W −1/2,2 duality (see the appendix of [BGP91]). This equality allowed them to extend the notion of winding number to the discontinuous part of W 1/2,2 .…”