Geodesics on SO(n) and a class of spherically symmetric maps as solutions to a nonlinear generalised harmonic map problem

Day, Stuart and Taheri, Ali (2017) Geodesics on SO(n) and a class of spherically symmetric maps as solutions to a nonlinear generalised harmonic map problem. Topological Methods in Nonlinear Analysis. ISSN 1230-3429 (Accepted)

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Abstract

In this paper we address questions on existence, multiplicity as well as qualitative features including rotational symmetry for certain classes of geometrically motivated maps serving as solutions to the nonlinear system

−div[ F‘(|x|,|∇u|²)∇u ] = F‘(|x|,|∇u|²)|∇u|²u in Xⁿ,
|u| = 1 in Xⁿ,
u = ϕ on ∂Xⁿ.

Here ϕ ∈ C ∞(∂X n , S n−1 ) is a suitable boundary map, F 0 is the derivative of F with respect to the second argument, u ∈ W1,p(X n , S n−1 ) for a fixed 1 < p < ∞ and X n = {x ∈ R n : a < |x| < b} is a generalised annulus. Of particular interest are spherical twists and whirls, where following [26], a spherical twist refers to a rotationally symmetric map of the form u : x 7→ Q(|x|)x|x| −1 with Q some suitable path in C ([a, b], SO(n)) and a whirl has a similar but more complex structure with only 2-plane symmetries. We establish the existence of an infinite family of such solutions and illustrate an interesting discrepancy between odd and even dimensions.

Item Type: Article
Schools and Departments: School of Mathematical and Physical Sciences > Physics and Astronomy
Subjects: Q Science > QA Mathematics
Depositing User: Billy Wichaidit
Date Deposited: 02 Nov 2017 10:33
Last Modified: 13 Apr 2018 11:46
URI: http://sro.sussex.ac.uk/id/eprint/70850

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