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- Title
Vortex sheet solutions for the Ginzburg–Landau system in cylinders: symmetry and global minimality.
- Authors
Ignat, Radu; Rus, Mircea
- Abstract
We consider the Ginzburg–Landau energy E ε for R M -valued maps defined in a cylinder shape domain B N × (0 , 1) n satisfying a degree-one vortex boundary condition on ∂ B N × (0 , 1) n in dimensions M ≥ N ≥ 2 and n ≥ 1 . The aim is to study the radial symmetry of global minimizers of this variational problem. We prove the following: if N ≥ 7 , then for every ε > 0 , there exists a unique global minimizer which is given by the non-escaping radially symmetric vortex sheet solution u ε (x , z) = (f ε (| x |) x | x | , 0 R M - N ) , ∀ x ∈ B N that is invariant in z ∈ (0 , 1) n . If 2 ≤ N ≤ 6 and M ≥ N + 1 , then the following dichotomy occurs between escaping and non-escaping solutions: there exists ε N > 0 such that if ε ∈ (0 , ε N) , then every global minimizer is an escaping radially symmetric vortex sheet solution of the form R u ~ ε where u ~ ε (x , z) = ( f ~ ε (| x |) x | x | , 0 R M - N - 1 , g ε (| x |)) is invariant in z-direction with g ε > 0 in (0, 1) and R ∈ O (M) is an orthogonal transformation keeping invariant the space R N × { 0 R M - N } ; if ε ≥ ε N , then the non-escaping radially symmetric vortex sheet solution u ε (x , z) = (f ε (| x |) x | x | , 0 R M - N ) , ∀ x ∈ B N , z ∈ (0 , 1) n is the unique global minimizer; moreover, there are no bounded escaping solutions in this case. We also discuss the problem of vortex sheet S M - 1 -valued harmonic maps.
- Subjects
CYLINDER (Shapes); HARMONIC maps; SYMMETRY; SET-valued maps
- Publication
Calculus of Variations & Partial Differential Equations, 2024, Vol 63, Issue 2, p1
- ISSN
0944-2669
- Publication type
Article
- DOI
10.1007/s00526-023-02628-x