Minimizing the Dirichlet energy over a space of measure preserving maps

Let $\Omega \subset \R^n$ be a bounded Lipschitz domain and consider the Dirichlet energy functional $$ {\Bbb F} [\u , \Omega] := \frac{1}{2} \int_\Omega |
abla \u (\x )|^2 \, d\x, $$ over the space of measure preserving maps $$ {\Cal A}(\Omega)=\{\u \in W^{1,2}(\Omega, \R^n) : \u |_{\partial \Omega} = \x , \ \det
abla \u = 1 \text{ ${\Cal L}^n$-a.e. in $\Omega$} \}. $$ Motivated by their significance in topology and the study of mapping class groups, in this paper we consider a class of maps, referred to as {\it twists}, and examine them in connection with the Euler--Lagrange equations associated with ${\Bbb F}$ over ${\Cal A}(\Omega)$. We investigate various qualitative properties of the resulting solutions in view of a remarkably simple, yet seemingly unknown explicit formula, when $n=2$.