Generalised twists, $\mathrm{\mathbf{SO}}(n)$, and the $p$-energy over a space of measure preserving maps

Abstract

Let Ω ⊂ R n be a bounded Lipschitz domain and consider the energy functionalIn this paper we introduce a class of maps referred to as generalised twists and examine them in connection with the Euler-Lagrange equations associated with F p over A p (Ω). The main result is a surprising discrepancy between even and odd dimensions. Here we show that in even dimensions the latter system of equations admit infinitely many smooth solutions, modulo isometries, amongst such maps. In odd dimensions this number reduces to one. The result relies on a careful analysis of the full versus the restricted Euler-Lagrange equations where a key ingredient is a necessary and sufficient condition for an associated vector field to be a gradient.

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

“…We see that the above has a unique solution for each m ∈ Z given explicitly by the formulation 20) where |ρ| 2 = N l=1 ρ 2 l and for the integral on the right a ≤ t ≤ b. Using the techniques developed and used in Section 2 one then proves the following statement.…”

“…See also [8,10,14] and the references therein.) The first class of maps we examine in this paper as solutions to the nonlinear system (1.7) are the so-called spherical twists as introduced in [26] (see also [20,21,24,25]). Recall that a spherical twist by definition is a map u ∈ C (X n , S n−1 ) of the form…”

Twist maps as energy minimisers in homotopy classes: Symmetrisation and the coarea formula

“…Indeed direct computations give the angle of rotation g = g(r) to be g(r) = 2πk log(r/a) log(b/a) + 2πm, k, m ∈ Z, (1.10) when n = 2 and g(r) = 2πk (r/a) 2−n − 1 (b/a) 2−n − 1 + 2πm, k, m ∈ Z, (1.11) when n ≥ 4 even. (See also [13], [16], [18] for complementing and further results.) Our point of departure is (1.4)-(1.9) and the aim is to study the minimising properties the twist maps calculated above.…”

“…[u; X] = n −n/|Qθ| 2 n/2 dx. (6.11)Now restricting to the particular case of the twist map being (cf [13]…”

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]).…”