Scaling Limits of the Chern-simons-higgs Energy

We continue our study in [16] of the Gamma limit of the Abelian Chern-Simons-Higgs energy Gcsh := 1 2 ∫ U |∇Aεuε| 2 + με 4 |curlAε − hex| |uε| + 1 ε2 |uε|(1− |uε|)dx on a bounded, simply connected, two dimensional domain where ε → 0 and με → μ ∈ [0,+∞]. Under the critical scaling, Gcsh ≈ |log ε|, 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 με → 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.