Spherical twists, stationary loops and harmonic maps from generalised annuli into spheres

Taheri, Ali (2011) Spherical twists, stationary loops and harmonic maps from generalised annuli into spheres. Nonlinear Differential Equations and Applications, 19 (1). pp. 79-95. ISSN 1021-9722

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Abstract

Let X ⊂ Rⁿ be a generalised annulus and consider the Dirichlet energy functional

E[u; X] := 1/2∫X |∇u(x)|²dx,

on the space of admissible maps

Aϕ(X) = u ∈ W²,¹ (X, Sⁿˉ¹) : u|∂X = ϕ . Here ϕ ∈ C(∂X, Sⁿˉ¹) is fixed and Aϕ(X) is non-empty. In this paper we introduce a class of maps referred to as spherical twists and examine them in connection with the Euler–Lagrange equation associated with E[·, X] on Aϕ(X) [the so-called harmonic map equation on X]. The main result here is an interesting discrepancy between even and odd dimensions. Indeed for even n subject to a compatibility condition on ϕ the latter system admits infinitely many smooth solutions modulo isometries whereas for odd n this number reduces to one or none. We discuss qualitative features of the solutions in view of their novel and explicit representation through the exponential map of the compact Lie group SO(n).

Item Type: Article
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Research Centres and Groups: Analysis and Partial Differential Equations Research Group
Depositing User: Ali Taheri
Date Deposited: 07 Nov 2017 10:20
Last Modified: 07 Nov 2017 10:20
URI: http://sro.sussex.ac.uk/id/eprint/70966

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