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

We consider the Gamma limit of the Abelian Chern–Simons–Higgs energyGcsh:=12∫U|∇Aεuε|2+μ24|curlAε−hex|2|uε|2+1ε2|uε|2(1−|uε|2)2dx on a bounded, simply connected, two-dimensional domain under the ε→0ε→0 limit. As a first step we study the Gamma limit ofEcsh:=12∫U|∇uε|2+1ε2|uε|2(1−|uε|2)2dx under two different scalings; Ecsh≈|logε|Ecsh≈|logε| and Ecsh≈|logε|2Ecsh≈|logε|2. We apply the |logε|2|logε|2-scaling result to the full Chern–Simons–Higgs energy GcshGcsh, and as a consequence we are able to compute the first critical field H1=H1(U,μ)H1=H1(U,μ) for the nucleation of a vortex. The method entails estimating in certain weak topologies the Jacobian J(uε)=det(∇uε)J(uε)=det(∇uε) in terms of the Chern–Simons–Higgs energy EcshEcsh.