Gamma limit of the nonself-dual Chern–Simons–Higgs energy

We consider the Gamma limit of the Abelian Chern–Simons–Higgs energy Gcsh := 1 2 U |∇Aεuε|2 + μ2 4 |curlAε − hex|2 |uε|2 + 1 ε2 |uε|2 1 − |uε|2 2 dx on a bounded, simply connected, two-dimensional domain under the ε→0 limit. As a first step we study the Gamma limit of Ecsh := 1 2 U |∇uε|2 + 1 ε2 |uε|2 1 − |uε|2 2 dx under two different scalings; Ecsh ≈ |log ε| and Ecsh ≈ |log ε|2. We apply the |log ε|2-scaling result to the full Chern–Simons–Higgs energy Gcsh, and as a consequence we are able to compute the first critical field H1 = H1(U,μ) for the nucleation of a vortex. The method entails estimating in certain weak topologies the Jacobian J(uε) = det(∇uε) in terms of the Chern–Simons–Higgs energy Ecsh. © 2008 Elsevier Inc. All rights reserved.