Stability and local minimality of spherical harmonic twists u = Q(|x|)x|x| −1 , positivity of second variations and conjugate points on SO(n)

Day, Stuart and Taheri, Ali (2020) Stability and local minimality of spherical harmonic twists u = Q(|x|)x|x| −1 , positivity of second variations and conjugate points on SO(n). The Journal of Analysis, 28 (2). pp. 431-460. ISSN 0971-3611

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Abstract

In this paper we discuss the stability and local minimising properties of spherical twists that arise as solutions to the harmonic map equation

HME[u; X n , S n−1 ] :=    ∆u + |∇u| 2 u = 0 in X n , |u| = 1 in X n , u = ϕ on ∂X n ,

by way of examining the positivity of the second variation of the associated Dirichlet energy. Here, following [30], by a spherical twist we mean a map u ∈ W 1,2 (X n , S n−1 ) of the form x 7→ Q(|x|)x|x| −1 where Q = Q(r) lies in C ([a, b], SO(n)) and X n = {x ∈ R n : a < |x| < b} (n ≥ 2). It is shown that subject to a structural condition on the twist path the energy at the associated spherical twist solution to the system has a positive definite second variation and subsequently proven to furnish a strong local energy minimiser. A detailed study of Jacobi fields and conjugate points along the twist path Q(r) = exp(G (r)H) and geodesics on SO(n) is undertaken and its remarkable implication and interplay on the minimality of spherical harmonic twists exploited.

Item Type: Article
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Research Centres and Groups: Analysis and Partial Differential Equations Research Group
Related URLs:
Depositing User: Ali Taheri
Date Deposited: 09 May 2019 14:05
Last Modified: 09 Jul 2020 13:45
URI: http://sro.sussex.ac.uk/id/eprint/83619

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