SCALING LIMITS OF THE CHERN–SIMONS–HIGGS ENERGY

Abstract

We continue our study in [16] of the Gamma limit of the Abelian Chern–Simons–Higgs energy [Formula: see text] on a bounded, simply connected, two-dimensional domain where ε → 0 and με → μ ∈ [0, +∞]. Under the critical scaling, Gcsh ≈ | log ε2, we establish the Gamma limit when μ ∈ (0,+∞], and as a consequence, we are able to compute the first critical field H1 = H1(U,μ) for the nucleation of a vortex. Finally, we show failure of Gamma convergence when μμ → 0 (this includes the self-dual case). The method entails estimating in certain weak topologies the Jacobian J(uε) = det (∇ uε) in terms of the Chern–Simons–Higgs energy Ecsh.