Generalised twists, stationary loops and the Dirichlet energy over a space of measure preserving maps

Shahrokhi-Dehkordi, M S and Taheri, A (2009) Generalised twists, stationary loops and the Dirichlet energy over a space of measure preserving maps. Calculus of Variations and Partial Differential Equations, 35 (2). pp. 191-213. ISSN 0944-2669

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Abstract

Let $${\Omega \subset \mathbb{R}^n}$$ be a bounded Lipschitz domain and consider the Dirichlet energy functional $${\mathbb F} [{\bf u}, \Omega] := \frac{1}{2} \int\limits_\Omega|\nabla {\bf u}({\bf x})|^2 \, d{\bf x},$$ over the space of measure preserving maps $${\mathcal A}(\Omega)=\left\{{\bf u}\in W^{1,2}(\Omega, \mathbb{R}^n) : {\bf u}|_{\partial \Omega} = {\bf x}, \mbox{ }\det \nabla {\bf u} = 1 \mbox{ }{{\rm a.e}.\; {\rm in} \Omega}\right\}.$$ In this paper we introduce a class of maps referred to as generalised twists and examine them in connection with the Euler¿Lagrange equations associated with $${{\mathbb F}}$$ over $${{\mathcal A}(\Omega)}$$ . The main result here is that in even dimensions the latter equations admit infinitely many solutions, modulo isometries, amongst such maps. We investigate various qualitative properties of these solutions in view of a remarkably interesting previously unknown explicit formula.

Item Type: Article SOME OF EQUATION LOST IN ABSTRACT School of Mathematical and Physical Sciences > Mathematics Mohammad Shahrokhi-Dehkordi 06 Feb 2012 20:14 18 Sep 2018 13:08 http://srodev.sussex.ac.uk/id/eprint/24881