2021/11/15 by Ignat, Radu, Nguyen, Luc · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2111.07669
We study the existence, uniqueness and minimality of critical points of the form mε,η(x) = (fε,η(|x|)(x)/(|x|), gε,η(|x|)) of the functional Eε,η[m] = ∫BN [(1)/(2) |∇ m|2 + (1)/(2ε2) (1 - |m|2)2 + (1)/(2η2) mN+12] dx for m=(m1, …, mN, mN+1) ∈ H1(BN,ℝN+1) with m(x) = (x,0) on ∂ BN. We establish a necessary and sufficient condition on the dimension N and the parameters ε and η for the existence of an escaping vortex solution (fε,η, gε,η) with gε,η> 0. We also establish its uniqueness and local minimality. In the limiting case η= 0, we prove the local minimality of the degree-one vortex solution for the Ginzburg-Landau (GL) energy for every ε > 0 and N ≥ 2. Similarly, when ε = 0, we prove the local minimality of the degree-one escaping vortex solution to an \mathbbSN-valued GL model arising in micromagnetics for every η> 0 and 2 ≤ N ≤ 6.