On the stability of the Ginzburg–Landau vortex

Abstract

We introduce a functional framework taylored to investigate the minimality and stability properties of the Ginzburg–Landau vortex of degree one on the whole plane. We prove that a renormalized Ginzburg–Landau energy is well defined in that framework and that the vortex is its unique global minimizer up to the invariances by translation and phase shift. Our main result is a nonlinear coercivity estimate for the renormalized energy around the vortex, from which we can deduce its orbital stability as a solution to the Gross–Pitaevskii equation, the natural Hamiltonian evolution equation associated to the Ginzburg–Landau energy.