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

journal contribution
posted on 2023-06-08, 05:34 authored by M S Shahrokhi-Dehkordi, Ali TaheriAli Taheri
Let $${\Omega \subset \mathbb{R}^n}$$ be a bounded Lipschitz domain and consider the Dirichlet energy functional $${\mathbb F} [{\bf u}, \Omega] := \frac{1}{2} \int\limits_\Omega|
abla {\bf u}({\bf x})|^2 \, d{\bf x},$$ over the space of measure preserving maps $${\mathcal A}(\Omega)=\left\{{\bf u}\in W^{1,2}(\Omega, \mathbb{R}^n) : {\bf u}|_{\partial \Omega} = {\bf x}, \mbox{ }\det
abla {\bf u} = 1 \mbox{ }{{\rm a.e}.\; {\rm in} \Omega}\right\}.$$ 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.

History

Publication status

  • Published

Journal

Calculus of Variations and Partial Differential Equations

ISSN

0944-2669

Publisher

Springer Verlag

Issue

2

Volume

35

Page range

191-213

Pages

13.0

Department affiliated with

  • Mathematics Publications

Notes

SOME OF EQUATION LOST IN ABSTRACT

Full text available

  • No

Peer reviewed?

  • Yes

Legacy Posted Date

2012-02-06

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