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

Abstract

Article (Accepted Version) http://sro.sussex.ac.uk Day, Stuart and Taheri, Ali (2018) 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, 51 (2). pp. 637-662.

A class of extremising sphere-valued maps with inherent maximal tori symmetries in SO ( n ) $\mathbf{SO}(n)$

“…Hence, with a slight abuse of notation, the associated spherical whirl has the form u = Q(r)x|x| -1 where r)H) where G = G (r) is as in Proposition 2.2 and H is the constant n × n skew-symmetric matrix from (2.12). It therefore follows from similar results in [2] (see also [3]) that the spherical whirl u = Q(ρ 1 , . .…”

Topology of twists, extremising twist paths and multiple solutions to the nonlinear system in variation $\mathscr{L}[u] = \nabla \mathscr{P}$

“…In surface topology the significance of twists [also known as Dehn twists] and their role as generators of the mapping class group of Riemannan surfaces has a long and rich history ( [6]). More recently in geometric analysis and PDEs these two dimensional twists and their higher dimensional counterparts (as above) have proven highly useful in establishing the existence of multiple solutions and multiple equilibria of different topological types (see [17,18,25], [31]- [34] as well as [10,11,23,24,28,27]).…”

On a class of stationary loops on SO ( n ) $\mathbf{SO}(n)$ and the existence of multiple twisting solutions to a nonlinear elliptic system subject to a hard incompressibility constraint

“…For the sake of clarity, note that by a [classical] solution we hereafter mean a pair (u, P) with u of class C 2 (U, R n ) ∩ C (U, R n ) and P of class C 1 (U) ∩ C (U) such that (1.2) holds in a pointwise sense in U. a Now, proceeding forward and arguing either formally and in a distributional sense, or classically, upon assuming further differentiability on L F , it is seen from (1.2)- Note, however, that this condition alone, unless U has a particular homology, does not imply that L F [u] is a gradient field in U, here, ∇P. For more on the background formulation and applications of system (1.2)-(1.3), in particular to function theory, mechanics, and nonlinear elasticity, see [2,3,5,10,14,19] and [1,4,7,11,12,[15][16][17][18] as well as [20][21][22][23][24][25][26][27]30] and [9,13,29,31,32] for related results and further applications.…”