Higher dimensional Ginzburg–Landau equations under weak anchoring boundary conditions

Abstract

For n≥3 and 0<ϵ≤1, let Ω⊂Rn be a bounded, simply connected, smooth domain, and uϵ:Ω⊂Rn→R2 solve the Ginzburg–Landau equation under the weak anchoring boundary condition:{−Δuϵ=1ϵ2(1−|uϵ|2)uϵinΩ,∂uϵ∂ν+λϵ(uϵ−gϵ)=0on∂Ω, where the anchoring strength parameter λϵ=Kϵ−α for some K>0 and α∈[0,1), and gϵ∈C2(∂Ω,S1). Motivated by the connection with the Landau–De Gennes model of nematic liquid crystals under weak anchoring conditions, we study the asymptotic behavior of uϵ as ϵ goes to zero under the condition that the total modified Ginzburg–Landau energy satisfies Fϵ(uϵ,Ω)≤M|log⁡ϵ| for some M>0.