Introduction

Abstract

This book is devoted to the mathematical study of the two-dimensional Ginzburg-Landau model with magnetic field. This is a model of great importance and recognition in physics (with several Nobel prizes awarded for it: Landau, Ginzburg, and Abrikosov). It was introduced by Ginzburg and Landau (see [101]) in the 1950s as a phenomenological model to describe superconductivity. Superconductivity was itself discovered in 1911 by Kammerling Ohnes. It consists in the complete loss of resistivity of certain metals and alloys at very low temperatures. The two most striking consequences of it are the possibility of permanent superconducting currents and the particular behavior that, when the material is submitted to an external magnetic field, that field gets expelled from it. Aside from explaining these phenomena, and through the very influential work of A. Abrikosov [1], the Ginzburg-Landau model allows one to predict the possibility of a mixed state in type II superconductors where triangular vortex lattices appear. These vortices—in a few words a vortex can be described as a quantized amount of vorticity of the superconducting current localized near a point—have since been the objects of many observations and experiments. The first observation dates back from 1967, by Essman and Trauble, see [93]. For pictures of lattice observations in superconductors and more references to experimental results, refer to the web page http://www.fys.uio.no/super/vortex/.