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

Let $\text{X}=\text{X}[a,b]=\{x:a<\mathrm{\mid }x\mathrm{\mid }<b\}\subset {\mathbb{R}}^n$ with $0<a<b<\mathrm{\infty }$ fixed be an open annulus and consider the energy functional, \begin{equation*} {\mathbb F} [u; \X] = \frac{1}{2} \int_\X \frac{|\nabla u|^2}{|u|^2} , dx, \end{equation*} over the space of admissible incompressible Sobolev maps \begin{equation*} {\mathcal A}_\phi(\X) = \bigg{ u \in W^{1,2}(\X, \R^n) : \det \nabla u = 1 $\mathit{\text{a.e.}}\text{ in }\text{X}\text{ and }u{\mathrm{\mid }}_{\mathrm{\partial }\text{X}}=\varphi$ \bigg}, \end{equation*} where $\varphi$ is the identity map of $\bar{\text{X}}$. Motivated by the earlier works \cite{TA2, TA3} in this paper we examine the {\it twist} maps as extremisers of $\mathbb{F}$ over ${\mathcal{A}}_{\varphi }(\text{X})$ and investigate their minimality properties by invoking the coarea formula and a symmetrisation argument. In the case $n=2$ where ${\mathcal{A}}_{\varphi }(\text{X})$ is a union of infinitely many disjoint homotopy classes we establish the minimality of these extremising twists in their respective homotopy classes a result that then leads to the latter twists being $L^1$-local minimisers of $\mathbb{F}$ in ${\mathcal{A}}_{\varphi }(\text{X})$. We discuss variants and extensions to higher dimensions as well as to related energy functionals.