### 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 $n=2$.

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 $n=2$.

#### Keywords

Dirichlet energy problem; variational methods; measure preserving maps

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