Local minimality of RN-valued and SN-valuedGinzburg–Landau vortex solutions in the unit ball BN

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] = Z
BN
h
1
2
|∇m|
2 +
1
2ε
2
(1 − |m|
2
)
2 +
1
2η
2 m2
N+1i
dx
for m = (m1, . . . , mN , mN+1) ∈ H1
(BN , R
N+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 particular, when 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 S
N -valued GL model in micromagnetics for all
η > 0 and 2 ≤ N ≤ 6.