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

Abstract

Let $\X = \X[a, b] = \{x: a<|x|<b\}\subset \R^n$ with $0<a<b<\infty$ 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}_ϕ(\X) = \bigg\{ u \in W^{1,2}(\X, \R^n) : \det \nabla u = 1 \text{ {\it a.e.} in $\X$ and $u|_{\partial \X} = ϕ$} \bigg\}, \end{equation*} where $ϕ$ is the identity map of $\overline \X$. Motivated by the earlier works \cite{TA2, TA3} in this paper we examine the {\it twist} maps as extremisers of ${\mathbb F}$ over ${\mathcal A}_ϕ(\X)$ and investigate their minimality properties by invoking the coarea formula and a symmetrisation argument. In the case $n=2$ where ${\mathcal A}_ϕ(\X)$ 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 ${\mathcal A}_ϕ(\X)$. We discuss variants and extensions to higher dimensions as well as to related energy functionals.