Minimizing the Dirichlet energy over a space of measure preserving maps

Abstract

Let $\Omega \subset \mathbb R^n$ be a bounded Lipschitz domain and
consider the Dirichlet energy functional
$$
{\mathbb F} [\u , \Omega] := \frac{1}{2} \int_\Omega |\nabla
\u (\x )|^2 dx,
$$
over the space of measure preserving maps
$$
{\mathcal A}(\Omega)=\{\u \in W^{1,2}(\Omega, \mathbb R^n) :
\u |_{\partial \Omega} = \x , \ \det \nabla \u = 1
\text{ ${\mathcal L}^n$-a.e in $\Omega$} \}.
$$
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 ${\mathbb F}$ over ${\mathcal A}(\Omega)$. We investigate
various qualitative properties of the resulting solutions in view
of a remarkably
simple, yet seemingly unknown explicit formula, when $n=2$.