The existence of multiple topologically distinct solutions to $\sigma_{2,p}$-energy

Abstract

Let ${\mathbb X} \subset \mathbb R^n$ be a bounded Lipschitz domain and consider the $\sigma_{2,p}$-energy functional\begin{equation*}{{\mathbb F}_{\sigma_{2,p}}}[u; {\mathbb X}] := \int_{\mathbb X} \big|{\wedge}^2 \nabla u\big|^p dx,\end{equation*}with $p\in ]1, \infty]$ over the space of measure preserving maps\begin{equation*}{\mathcal A}_p(\mathbb X) =\big\{u \in W^{1,2p}\big(\mathbb X, \mathbb R^n\big) : u|_{\partial \mathbb X} = {x},\ \det \nabla u =1\mbox{ for ${\mathcal L}^n$-a.e. in $\mathbb X$} \big\}.\end{equation*}In this article we address the question of multiplicity versus uniqueness for extremals and strong local minimizers of the $\sigma_{2,p}$-energy funcional $\mathbb F_{\sigma_{2,p}}[\cdot; {\mathbb X}]$ in ${\mathcal A}_p({\mathbb X})$. We use a topological class of maps referred to as generalised twists and examine them in connection with the Euler-Lagrange equations associated with $\sigma_{2,p}$-energy functional over ${\mathcal A}_p({\mathbb X})$. Most notably, we prove the existence of a countably infinite of topologically distinct twisting solutions to the later system in all even dimensions by linking the system to a set of nonlinear isotropic ODEs on the Lie group ${\rm SO}(n)$. In sharp contrast in odd dimensions the only solution is the map $u\equiv x$. The result relies on a careful analysis of the full versus the restricted Euler-Lagrange equations. Indeed, an analysis of curl-free vector fields generated by symmetric matrix fields plays a pivotal role.