We study the asymptotic behavior of a family of functional describing the formation of topologically induced boundary vortices in thin magnetic films. We obtain convergence results for sequences of minimizers and some classes of stationary points, and relate the limiting behavior to a finite dimensional problem, the renormalized energy associated to the vortices.
Abstract. For a one-dimensional nonlocal nonconvex singular perturbation problem with a noncoercive periodic well potential, we prove a Γ-convergence theorem and show compactness up to translation in all L p and the optimal Orlicz space for sequences of bounded energy. This generalizes work of Alberti, Bouchitté and Seppecher (1994) for the coercive two-well case. The theorem has applications to a certain thin-film limit of the micromagnetic energy.Mathematics Subject Classification. 49J45.
We consider a complex Ginzburg-Landau equation that contains a Schrödinger term and a damping term that is proportional to the time derivative. Given well-prepared initial conditions that correspond to quantized vortices, we establish the vortex motion law until collision time.
We consider the Gamma limit of the Abelian Chern-Simons-Higgs energyon a bounded, simply connected, two-dimensional domain under the ε → 0 limit. As a first step we study the Gamma limit ofunder two different scalings; E csh ≈ |log ε| and E csh ≈ |log ε| 2 . We apply the |log ε| 2 -scaling result to the full Chern-Simons-Higgs energy G csh , 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. The method entails estimating in certain weak topologies the Jacobian J (u ε ) = det(∇u ε ) in terms of the Chern-Simons-Higgs energy E csh .
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