Two results on the equivariant Ginzburg–Landau vortex in arbitrary dimension

Abstract

We characterize the O(N)-equivariant vortex solution for Ginzburg-Landau type equations in the Ndimensional Euclidean space and we prove its local energy minimality for the corresponding energy functional.

Uniaxial versus biaxial character of nematic equilibria in three dimensions

“…In the GL-framework, researchers study maps, u : R d → R m , d, m = 2, 3 (see e.g. [5,25,31]), whereas the LdG variable is a five-dimensional map, Q : R 3 → R 5 . A uniaxial Q-tensor has two degenerate non-zero eigenvalues and hence, three degrees of freedom and in this case, there is broader scope for transferable methodologies (see for example, [15]).…”

Symmetry in nonlinear PDEs: Some open problems

“…More recently (and quite surprisingly), it was shown in [24] that the answer is negative for N = 8, due to the existence of a degree-one harmonic eigenmap on S 7 , but energy 310 A. Pisante JFPTA minimality of this solution is unknown. On the other hand, local energy minimality of (3.7) is known in any dimension (see [51]). Thus, we will not discuss the previous question in more details, but we instead restrict to local energy minimizers of (3.4) and we consider the following problem.…”

“…In view of the above principles, the proof of the symmetry result relies on a detailed study of the minimizing solution u at infinity, i.e., on the analysis of the scaled maps u R (x) = u(Rx) as R → ∞, a suitable classification of their possible limits and a way to propagate the symmetry of the latter usually based on integral identities (for more details on this last step we refer the interested reader to [42,44,51]). …”

“…Thus, the main difficulty at present seems to be the lack of a corresponding rigidity result for the degree-zero homogeneous locally minimizing harmonic map in H 1 loc (R N ; S N −1 ) for N ≥ 4. Finally, motivated by the results in [51], we discuss a similar symmetry problem on a bounded domain. Let N ≥ 3 and let B ⊂ R N be the unit ball.…”

“…First note that for any ε > 0, equation (3.3) admits an O(N )-equivariant solution in a form U ε analogous to (3.7) with boundary value U ε (x) = x |x| on ∂B. On the other hand, the corresponding energy minimizers u ε are known to satisfy [51,Proposition 3.3]). In particular, for ε > 0 small enough, each minimizer u ε has a single isolated zero whose distance from the origin tends to zero in the limit.…”