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 $n=2$.