On the first critical field of a Ginzburg-Landau type problem for anisotropic superconductors
Superconductivity is an important phenomenon in condensed matter physics. There have been significant efforts devoted to understanding this phenomenon, and different physical models have been proposed to describe and predict behaviors of superconductors. The celebrated Ginzburg-Landau model has been well accepted as a macroscopic model for isotropic superconductors. However, for a large class of high temperature superconductors, a common feature is the discrete layered structure which makes these materials highly anisotropic and thus cannot be described appropriately by the Ginzburg-Landau model (a continuous model). In this talk, we will discuss the Lawrence-Doniach model which is a modification of the Ginzburg-Landau model to account for the discrete layered structure. One of the central questions in the mathematical study of such problems is to understand the defect structure in the material subject to magnetic fields of different strengths. We will present some recent results on the first critical field corresponding to a phase transition in the material from the superconducting state to the mixed state (coexistence of superconducting and normal states). This is a delicate physical regime which requires comprehensive machineries in the analysis.