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- Title
THE EXISTENCE OF MULTIPLE TOPOLOGICALLY DISTINCT SOLUTIONS TO σ<sub>2;p</sub>-ENERGY.
- Authors
TAGHAVI, MOJGAN; SHAHROKHI-DEHKORDI, MOHAMMAD S.
- Abstract
Let 핏 ⊂ ℝn be a bounded Lipschitz domain and consider the σ2,p-energy functional with 픽 σ2,p[u; 핏] := ∫ 핏| Λ² ∇ u|pdx, with p ∈] 1, ∞] over the space of measure preserving maps Ap(핏) = {u ∈ W1,2p(핏, ℝn) : u|∂핏 = x, det ∇u = 1 for Ln-a.e. in 핏}. In this article we address the question of multiplicity versus uniqueness for extremals and strong local minimizers of the σ2,p-energy funcional σσ2,p [ •;핏] in Ap(핏). We use a topological class of maps referred to as generalised twists and examine them in connection with the Euler--Lagrange equations associated with σ2,p-energy functional over Ap(핏). Most notably, we prove the existence of a countably infinite of topologically distinct twisting solutions to the later system in all even dimensions by linking the system to a set of nonlinear isotropic ODEs on the Lie group SO(n). In sharp contrast in odd dimensions the only solution is the map u = x. The result relies on a careful analysis of the full versus the restricted Euler-- Lagrange equations. Indeed, an analysis of curl-free vector fields generated by symmetric matrix fields plays a pivotal role.
- Subjects
LAGRANGE equations; VECTOR fields; VECTOR analysis; SYMMETRIC matrices; SKYRME model; LIE groups
- Publication
Topological Methods in Nonlinear Analysis, 2023, Vol 62, Issue 2, p409
- ISSN
1230-3429
- Publication type
Article
- DOI
10.12775/TMNA.2023.010