Generalised twists, stationary loops and the Dirichlet energy over a space of measure preserving maps

Let $${\Omega \subset \mathbb{R}^n}$$ be a bounded Lipschitz domain and consider the Dirichlet energy functional $${\mathbb F} [{\bf u}, \Omega] := \frac{1}{2} \int\limits_\Omega|\nabla {\bf u}({\bf x})|^2 \, d{\bf x},$$ over the space of measure preserving maps $${\mathcal A}(\Omega)=\left\{{\bf u}\in W^{1,2}(\Omega, \mathbb{R}^n) : {\bf u}|_{\partial \Omega} = {\bf x}, \mbox{ }\det \nabla {\bf u} = 1 \mbox{ }{{\rm a.e}.\; {\rm in} \Omega}\right\}.$$ In 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 $${{\mathbb F}}$$ over $${{\mathcal A}(\Omega)}$$ . The main result here is that in even dimensions the latter equations admit infinitely many solutions, modulo isometries, amongst such maps. We investigate various qualitative properties of these solutions in view of a remarkably interesting previously unknown explicit formula