# Minimizers of the magnetic Ginzburg-Landau functional in simply connected domain with prescribed degree on the boundary

Leonid Berlyand, Oleksandr Misiats, Volodymyr Rybalko

Research output: Contribution to journalArticle

4 Scopus citations

### Abstract

We study the minimizers of the GinzburgLandau free energy functional in the class (u, A) ∈ H1(Ω; ) × H1(Ω; 2) with |u| = 1 on ∂Ω, where Ω is a bounded simply connected domain in 2. We consider the connected components of this class defined by the prescribed topological degree d of u on the boundary ∂Ω. We show that for d ≠ 0 the minimizers exist if 0 < λ ≤ 1 and do not exist if λ > 1, where λ is the coupling constant ($\sqrt{\lambda/2}$ is the GinzburgLandau parameter). We also establish the asymptotic locations of vortices for λ → 1 - 0 (the critical value λ = 1 is known as the Bogomol'nyi integrable case).

Original language English (US) 53-66 14 Communications in Contemporary Mathematics 13 1 https://doi.org/10.1142/S0219199711004130 Published - Feb 2011

### All Science Journal Classification (ASJC) codes

• Mathematics(all)
• Applied Mathematics