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 $$ {\mathcal A}(\Omega)=\{u \in W^{1,2}(\Omega, \mathbb R^n) : u |_{\partial \Omega} = x , {\rm det}\, \nabla u = 1 \text{ ${\mathcal L}^n$-a.e. in $\Omega$} \}. $$ 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 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 $n=2$.