Near boundary vortices in a magnetic Ginzburg–Landau model: Their locations via tight energy bounds

Abstract

Given a bounded doubly connected domain G ⊂ R 2 , we consider a minimization problem for the Ginzburg–Landau energy functional when the order parameter is constrained to take S 1 -values on ∂ G and have degrees zero and one on the inner and outer connected components of ∂ G, correspondingly. We show that minimizers always exist for 0 < λ < 1 and never exist for λ ⩾ 1 , where λ is the coupling constant ( λ / 2 is the Ginzburg–Landau parameter). When λ → 1 − 0 minimizers develop vortices located near the boundary, this results in the limiting currents with δ-like singularities on the boundary. We identify the limiting positions of vortices (that correspond to the singularities of the limiting currents) by deriving tight upper and lower energy bounds. The key ingredient of our approach is the study of various terms in the Bogomol'nyi's representation of the energy functional.