The Existence of Radially Symmetric Vortices in a Ferromagnetic Model of Superconductivity

Abstract

We take a model for Ferromagnetic Superconductors based on a variational energy functional, and search for radially symmetric minimizers. First we define what it means for a solution to the Euler-Lagrange equations to be admissible, before relating these admissible solutions to an appropriate function space. We then use a variational approach to prove the existence of minimizers. Since it is not clear at first whether or not the energy is bounded below, the direct method of the calculus of variations does not apply. Instead, we first prove existence in a case where the energy is bounded below, namely when the Zeeman coupling constant g vanishes. We then use the implicit function theorem to prove the existence of physically relevant minimizers for small values of g.