We continue our study in  of the Gamma limit of the Abelian ChernSimonsHiggs energy Gcsh := 1/2 ∫U |∇Aε uε|2 + με2/4 |curl Aε - h ex|2/|uε|2 + 1/ε2 |uε|2 ( 1 - |u ε|2)2 dx on a bounded, simply connected, two-dimensional domain where ε → 0 and μ(ε) → μ ∈ [0, +∞]. Under the critical scaling, G(csh) ≈ |log ε|2 , we establish the Gamma limit when μ ∈ (0,+∞], and as a consequence, we are able to compute the first critical field H"1 = H"1(U,μ) for the nucleation of a vortex. Finally, we show failure of Gamma convergence when μ(μ) → 0 (this includes the self-dual case). The method entails estimating in certain weak topologies the Jacobian J(u(ε)) = det(∇ u(ε)) in terms of the ChernSimonsHiggs energy E(csh).
Bibliographical noteFunding Information:
D. Spirn was supported in part by NSF grant DMS–0510121.
- Chern-Simons-Higgs theory