# Minimizing the Dirichlet energy over a space of measure preserving maps

Taheri, Ali (2009) Minimizing the Dirichlet energy over a space of measure preserving maps. Topological Methods in Nonlinear Analysis, 33 (1). pp. 179-204. ISSN 1230-3429

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## Abstract

Let $\Omega \subset \R^n$ be a bounded Lipschitz domain and consider the Dirichlet energy functional $${\Bbb F} [\u , \Omega] := \frac{1}{2} \int_\Omega |\nabla \u (\x )|^2 \, d\x,$$ over the space of measure preserving maps $${\Cal A}(\Omega)=\{\u \in W^{1,2}(\Omega, \R^n) : \u |_{\partial \Omega} = \x , \ \det \nabla \u = 1 \text{ {\Cal L}^n-a.e. in \Omega} \}.$$ 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 $n=2$.

Item Type: Article School of Mathematical and Physical Sciences > Mathematics Ali Taheri 06 Feb 2012 18:11 18 Sep 2018 13:09 http://sro.sussex.ac.uk/id/eprint/15125