Generalised twists,stationary loops,and the Dirichlet energy over a space of measure preserving maps期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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

Authors:

M S Shahrokhi-Dehkordi A Taheri

Institution:

(1) Department of Mathematics, University of Sussex, Falmer, Brighton, BN1 9RF, UK

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.

Keywords:

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